52 results on '"Roe Goodman"'
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2. Representation Theory and Analysis on Homogeneous Spaces
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Simon Gindikin, Roe Goodman, Frederick P. Greenleaf, Paul J. Sally, Jr, Simon Gindikin, Roe Goodman, Frederick P. Greenleaf, and Paul J. Sally, Jr
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- Nilpotent Lie groups--Congresses, p-adic groups--Congresses, Representations of groups--Congresses
- Abstract
Combining presentation of new results with in-depth surveys of recent work, this book focuses on representation theory and harmonic analysis on real and $p$-adic groups. The papers are based on lectures presented at a conference dedicated to the memory of Larry Corwin and held at Rutgers University in February 1993. The book presents a survey of harmonic analysis on nilpotent homogeneous spaces, results on multiplicity formulas for induced representations, new methods for constructing unitary representations of real reductive groups, and a unified treatment of trace Paley-Wiener theorems for real and $p$-adic reductive groups. In the representation theory of the general linear group over $p$-adic fields, the book provides a description of Corwin's contributions, a survey of the role of Hecke algebras, and a presentation of the theory of simple types. Other types of reductive $p$-adic groups are also discussed. Among the other topics included are the representation theory of discrete rational nilpotent groups, skew-fields associated to quadratic algebras, and finite models for percolation. A timely publication featuring contributions by some of the top researchers in the field, this book offers a perspective not often found in conference proceedings. more...
- Published
- 2011
Catalog
3. Quadratic algebras and skew-fields
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L. J. Corwin, I. M. Gel′fand, and Roe Goodman
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- 1994
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4. Lawrence J. Corwin (1943–1992)
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Simon Gindikin, Roe Goodman, Frederick P. Greenleaf, and Paul J. Sally
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- 1994
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5. Mathematics, Music, Masters
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Roe Goodman
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Musicology ,Mathematics education ,Music education - Published
- 2010
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6. Lie Groups and Algebraic Groups
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Nolan R. Wallach and Roe Goodman
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Classical group ,Pure mathematics ,Representation of a Lie group ,Group of Lie type ,Simple Lie group ,Real form ,Lie group ,Lie theory ,Group theory ,Mathematics - Abstract
Hermann Weyl, in his famous book The Classical Groups, Their Invariants and Representations [164], coined the name classical groups for certain families of matrix groups. In this chapter we introduce these groups and develop the basic ideas of Lie groups, Lie algebras, and linear algebraic groups. We show how to put a Lie group structure on a closed subgroup of the general linear group and determine the Lie algebras of the classical groups. We develop the theory of complex linear algebraic groups far enough to obtain the basic results on their Lie algebras, rational representations, and Jordan–Chevalley decompositions (we defer the deeper results about algebraic groups to Chapter 11). We show that linear algebraic groups are Lie groups, introduce the notion of a real form of an algebraic group (considered as a Lie group), and show how the classical groups introduced at the beginning of the chapter appear as real forms of linear algebraic groups. more...
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- 2009
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7. Branching Laws
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Roe Goodman and Nolan R. Wallach
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- 2009
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8. Tensor Representations of O(V) and Sp(V)
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Nolan R. Wallach and Roe Goodman
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Pure mathematics ,Symplectic group ,Symmetric group ,Irreducible representation ,Orthogonal group ,Tensor ,Representation theory ,Brauer algebra ,Mathematics ,Symplectic geometry - Abstract
In this chapter we analyze the action of the orthogonal and symplectic groups on the tensor powers of their defining representations. We show (following ideas of Weyl [164]) that the subspaces of harmonic tensors can be decomposed using the theory of Young symmetrizers from Chapter 9. This furnishes models (Weyl modules) for all the irreducible representations of the orthogonal and symplectic groups as spaces of harmonic tensors in the image of Young symmetrizers. Our approach involves the interplay of the commuting algebra (a quotient of the Brauer algebra) with the representation theory of the orthogonal and symplectic groups. The key observation is that the action of the Brauer algebra on the space of harmonic tensors factors through the action of the symmetric group on tensors. more...
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- 2009
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9. Classical Invariant Theory
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Roe Goodman and Nolan R. Wallach
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Pure mathematics ,Tensor product ,Invariant polynomial ,Irreducible representation ,Regular representation ,Geometric invariant theory ,Invariant (mathematics) ,Invariant theory ,Finite type invariant ,Mathematics - Abstract
For a linear algebraic group G and a regular representation (ρ,V) of G, the basic problem of invariant theory is to describe the G-invariant elements (⊗ k V) G of the k-fold tensor product for all k. If G is a reductive, then a solution to this problem for (ρ *, V *) leads to a determination of the polynomial invariants P(V) G . When G ⊂ GL(W) is a classical group and V = W k ⊕ (W *) l (k copies of W and l copies of W *), explicit and elegant solutions to the basic problem of invariant theory, known as the first fundamental theorem (FFT) of invariant theory for G, were found by Schur, Weyl, Brauer, and others. The fundamental case is G = GL(V) acting on V . Following Schur and Weyl, we turn the problem of finding tensor invariants into the problem of finding the operators commuting with the action of GL(V) on ⊗ k V, which we solved in Chapter 4. This gives an FFT for GL(V) in terms of complete contractions of vectors and covectors. When G is the orthogonal or symplectic group we first find all polynomial invariants of at most dim V vectors. We then use this special case to transform the general problem of tensor invariants for an arbitrary number of vectors into an invariant problem for GL(V) of the type we have already solved. more...
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- 2009
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10. Highest-Weight Theory
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Nolan R. Wallach and Roe Goodman
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Classical group ,Pure mathematics ,Weyl group ,symbols.namesake ,Lie algebra ,Adjoint representation ,symbols ,Maximal torus ,Unitarian trick ,Semisimple Lie algebra ,Casimir element ,Mathematics - Abstract
In this chapter we study the regular representations of a classical group G by the same method used for the adjoint representation. When G is a connected classical group, an irreducible regular G-module decomposes into a direct sum of weight spaces relative to the action of a maximal torus of G. The theorem of the highest weight asserts that among the weights that occur in the decomposition, there is a unique maximal element, relative to a partial order coming from a choice of positive roots for G. We prove that every dominant integral weight of a semisimple Lie algebra g is the highest weight of an irreducible finite-dimensional representation of g. When g is the Lie algebra of a classical group G, the corresponding regular representations of G are constructed in Chapters 5 and 6 and studied in greater detail in Chapters 9 and 10. A crucial property of a classical group is the complete reducibility of its regular representations. We give two (independent) proofs of this: one algebraic using the Casimir operator, and one analytic using Weyl’s unitarian trick. more...
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- 2009
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11. Representations on Spaces of Regular Functions
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Nolan R. Wallach and Roe Goodman
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Classical group ,Linear algebraic group ,Pure mathematics ,Borel subgroup ,Symmetric space ,Algebraic group ,Real form ,Lie group ,Affine variety ,Mathematics - Abstract
If G is a reductive linear algebraic group acting on an affine variety X, then G acts linearly on the function space ט[X]. In this chapter we will give several of the high points in the study of this representation. We will first analyze cases in which the representation decomposes into distinct irreducible representations (one calls X multiplicity-free in this case), give the most important class of such spaces (symmetric spaces), and determine the decomposition of ט[X] as a G-module in this case. We also obtain the second fundamental theorem of invariants for the classical groups from this approach. The philosophy in this chapter is that the geometry of the orbits of G in X gives important information about the structure of the corresponding representation of G on ט[X]. This philosophy is most apparent in the last part of this chapter, in which we give a new proof of a celebrated theorem of Kostant and Rallis concerning the isotropy representation of a symmetric space. This chapter is also less self-contained than the earlier ones. For example, the basic results of Chevalley on invariants corresponding to symmetric pairs are only quoted (although for the pairs of classical type these facts are verified on a case-by-case basis). We also mix algebraic and analytic techniques by viewing G both as an algebraic group and as a Lie group with a compact real form. more...
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- 2009
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12. Character Formulas
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Roe Goodman and Nolan R. Wallach
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- 2009
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13. Structure of Classical Groups
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Roe Goodman and Nolan R. Wallach
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Classical group ,Pure mathematics ,Direct sum ,Irreducible representation ,Lie algebra ,Lie group ,Maximal torus ,Killing form ,Semisimple Lie algebra ,Mathematics - Abstract
In this chapter we study the structure of a classical group G and its Lie algebra. We choose a matrix realization of G such that the diagonal subgroup H ⊂ G is a maximal torus; by elementary linear algebra every conjugacy class of semisimple elements intersects H. Using the unipotent elements in G, we show that the groups GL(n,\(\mathbb{C}\)), SL(n,\(\mathbb{C}\)), SO(n,\(\mathbb{C}\)), and Sp(n,\(\mathbb{C}\)) are connected (as Lie groups and as algebraic groups). We examine the group SL(2,C), find its irreducible representations, and show that every regular representation decomposes as the direct sum of irreducible representations. This group and its Lie algebra play a basic role in the structure of the other classical groups and Lie algebras. We decompose the Lie algebra of a classical group under the adjoint action of a maximal torus and find the invariant subspaces (called root spaces) and the corresponding characters (called roots). The commutation relations of the root spaces are encoded by the set of roots; we use this information to prove that the classical (trace-zero) Lie algebras are simple (or semisimple). In the final section of the chapter we develop some general Lie algebra methods (solvable Lie algebras, Killing form) and show that every semisimple Lie algebra has a root-space decomposition with the same properties as those of the classical Lie algebras. more...
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- 2009
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14. Algebraic Groups and Homogeneous Spaces
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Nolan R. Wallach and Roe Goodman
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Pure mathematics ,Real algebraic geometry ,Computer Science::Programming Languages ,Dimension of an algebraic variety ,Geometric invariant theory ,Representation theory ,Invariant theory ,Group theory ,Algebraic geometry and analytic geometry ,Moduli space ,Mathematics - Abstract
We now develop the theory of linear algebraic groups and their homogeneous spaces, as a preparation for the geometric approach to representations and invariant theory in Chapter 12.
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- 2009
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15. Tensor Representations of GL(V)
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Roe Goodman and Nolan R. Wallach
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Pure mathematics ,Tensor product ,Weyl module ,Irreducible representation ,Tensor (intrinsic definition) ,Duality (optimization) ,Reciprocity law ,Group algebra ,Mathematics::Representation Theory ,Invariant theory ,Mathematics - Abstract
In this chapter we bring together the representation theories of the groups GL(n,C) and б k via their mutually commuting actions on ⊗ k C n . We already exploited this connection in Chapter 5 to obtain the first fundamental theorem of invariant theory for GL(n, C). In this chapter we obtain the full isotypic decomposition of ⊗ k C n under the action of GL(n,C)× б k . This decomposition gives the Schur–Weyl duality pairing between the irreducible representations of GL(n,C) and those of б k . From this pairing we obtain the celebrated Frobenius character formula for the irreducible representations of б k . We then reexamine Schur–Weyl duality and GL(k,C)–GL(n,C) duality from Chapters 4 and 5 in the framework of dual pairs of reductive groups. Using the notion of seesaw pairs of subgroups, we obtain reciprocity laws for tensor products and induced representations. In particular, we show that every irreducible б k -module can be realized as the weight space for the character x ↦ det(x) in an irreducible GL(k,C) representation. Explicit models (the Weyl modules) for all the irreducible representations of GL(n,C) are obtained using Young symmetrizers. These elements of the group algebra of C[б k ] act as projection operators onto GL(n,C)-irreducible invariant subspaces. The chapter concludes with the Littlewood–Richardson rule for calculating the multiplicities in tensor products. more...
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- 2009
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16. Algebras and Representations
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Nolan R. Wallach and Roe Goodman
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Pure mathematics ,Finite group ,Interior algebra ,p-adic Hodge theory ,Algebraic group ,Algebra representation ,Group algebra ,Hopf algebra ,Invariant theory ,Mathematics - Abstract
In this chapter we develop some algebraic tools needed for the general theory of representations and invariants. The central result is a duality theorem for locally regular representations of a reductive algebraic group G. The duality between the irreducible regular representations of G and irreducible representations of the commuting algebra of G plays a fundamental role in classical invariant theory. We study the representations of a finite group through its group algebra and characters, and we construct induced representations and calculate their characters. more...
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- 2009
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17. Spinors
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Roe Goodman and Nolan R. Wallach
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- 2009
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18. Harmonic analysis on compact symmetric spaces
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Roe Goodman
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Algebra ,Harmonic analysis ,Pure mathematics ,Noncommutative harmonic analysis ,Algebra over a field ,Mathematics - Published
- 2008
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19. Book Review: Operators and representation theory: Canonical models for algebras of operators arising in quantum mechanics
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Roe Goodman
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Pure mathematics ,Operator algebra ,Applied Mathematics ,General Mathematics ,Mathematical formulation of quantum mechanics ,Method of quantum characteristics ,Creation and annihilation operators ,Symmetry in quantum mechanics ,Operator theory ,Second quantization ,Canonical commutation relation ,Mathematics - Published
- 1990
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20. Whittaker transforms on real-rank one Lie groups
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Roe Goodman
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Pure mathematics ,General Mathematics ,Simple Lie group ,Rank (graph theory) ,Lie group ,Lie theory ,Whittaker function ,Mathematics - Published
- 1990
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21. Multiplicity-Free Spaces and Schur–Weyl–Howe Duality
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Roe Goodman
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Pure mathematics ,Topological tensor product ,Multiplicity (mathematics) ,Mathematics - Published
- 2004
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22. Symmetry, Representations, and Invariants
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Roe Goodman, Nolan R. Wallach, Roe Goodman, and Nolan R. Wallach
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- Representations of groups, Invariants, Symmetry (Mathematics), Lie groups, Algebra, Darstellungstheorie, Invariante, Lineare algebraische Gruppe
- Abstract
Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications. The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case. Key Features of Symmetry, Representations, and Invariants: (1) Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus; (2) Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux); (3) Self-contained chapters, appendices, comprehensive bibliography; (4) More than 350 exercises (most with detailed hints for solutions) further explore main concepts; (5) Serves as an excellent main text for a one-year course in Lie group theory; (6) Benefits physicists as well as mathematicians as a reference work. more...
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- 2009
23. The algebra of K-invariant vector fields on a symmetric space G/K
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Ilka Agricola and Roe Goodman
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20G05 ,General Mathematics ,Adjoint representation ,Algebra ,Kernel (algebra) ,43A90 ,Symmetric space ,Algebraic group ,Simply connected space ,Lie algebra ,15A72 ,FOS: Mathematics ,53C35, 17B66, 58J70 ,22E46 ,Invariant (mathematics) ,Representation Theory (math.RT) ,Invariant differential operator ,Mathematics - Representation Theory ,14M17 ,Mathematics - Abstract
When $G$ is a complex reductive algebraic group and $G/K$ is a reductive symmetric space, the decomposition of $\C[G/K]$ as a $K$-module was obtained (in a non-constructive way) by Richardson, generalizing the celebrated result of Kostant-Rallis for the linearized problem (the harmonic decomposition of the isotropy representation). To obtain a constructive version of Richardson's results, this paper studies the infinite dimensional Lie algebra $\X(G/K)^K$ of $K$-invariant regular algebraic vector fields using the geometry of $G/K$ and the $K$-spherical representations of $G$. Assume $G$ is semisimple and simply-connected and let $\J$ be the algebra of $K$ biinvariant functions on $G$. An explicit set of free generators for the localization $ \X(G/K)^K_{\psi}$ is constructed for a suitable $\psi \in \J$. A commutator formula is obtained for $K$-invariant vector fields in terms of the corresponding $K$-covariant maps from $G$ to the isotropy representation of $G/K$. Vector fields on $G/K$ whose horizontal lifts to $G$ are tangent to the Cartan embedding of $G/K$ into $G$ are called \emph{flat}. When $G$ is simple and simply connected, it is shown that every element of $\X(G/K)^K$ is flat if and only if $K$ is semisimple. The gradients of the fundamental characters of $G$ are shown to generate all conjugation-invariant vector fields on $G$. These results are applied in the case of the adjoint representation of $G = \SL(2,\C)$ to construct a conjugation invariant differential operator whose kernel furnishes a harmonic decomposition of $\C[G]$., Comment: Latex2e, 18 pages more...
- Published
- 2003
24. Book Review: Hardy spaces on homogeneous groups
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Roe Goodman
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Pure mathematics ,symbols.namesake ,Homogeneous ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,symbols ,Hardy space ,Mathematics - Published
- 1983
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25. Higher-order Sugawara operators for affine Lie algebras
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Nolan R. Wallach and Roe Goodman
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Algebra ,Adjoint representation of a Lie algebra ,Affine representation ,Applied Mathematics ,General Mathematics ,Current algebra ,Killing form ,Kac–Moody algebra ,Affine Lie algebra ,Generalized Kac–Moody algebra ,Lie conformal algebra ,Mathematics - Abstract
Let g ^ \hat {\mathfrak {g}} be the affine Lie algebra associated to a simple Lie algebra g \mathfrak {g} . Representations of g ^ \hat {\mathfrak {g}} are described by current fields X ( ζ ) X(\zeta ) on the circle T ( X ∈ g {\mathbf {T}}\;(X \in \mathfrak {g} and ζ ∈ T ) \zeta \in {\mathbf {T}}) . In this paper a linear map σ \sigma from the symmetric algebra S ( g ) S(\mathfrak {g}) to (formal) operator fields on a suitable category of g ^ \hat {\mathfrak {g}} modules is constructed. The operator fields corresponding to g \mathfrak {g} -invariant elements of S ( g ) S(\mathfrak {g}) are called Sugawara fields. It is proved that they satisfy commutation relations of the form ( ∗ ) (\ast ) \[ [ σ ( u ) ( ζ ) , X ( η ) ] = c ∞ D δ ( ζ / η ) σ ( ∇ X u ) ( ζ ) + higher-order terms [\sigma (u)(\zeta ),X(\eta )] = {c_\infty }D\delta (\zeta /\eta )\sigma ({\nabla _X}u)(\zeta ) + {\text {higher-order}}\;{\text {terms}} \] with the current fields, where c ∞ {c_\infty } is a renormalization of the central element in g ^ \hat {\mathfrak {g}} and D δ D\delta is the derivative of the Dirac delta function. The higher-order terms in ( ∗ ) (\ast ) are studied using results from invariant theory and finite-dimensional representation theory of g \mathfrak {g} . For suitably normalized invariants u u of degree 4 4 or less, these terms are shown to be zero. This vanishing is also proved for g = sl ( n , C ) \mathfrak {g} = {\text {sl}}(n,{\mathbf {C}}) and u u running over a particular choice of generators for the symmetric invariants. The Sugawara fields defined by such invariants commute with the current fields whenever c ∞ {c_\infty } is represented by zero. This property is used to obtain the commuting ring, composition series, and character formulas for a class of highest-weight modules for g ^ \hat {\mathfrak {g}} . more...
- Published
- 1989
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26. Classical and quantum mechanical systems of Toda-lattice type
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Nolan R. Wallach and Roe Goodman
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Quantum dynamics ,35J10 ,Statistical and Nonlinear Physics ,Eigenfunction ,Type (model theory) ,58F05 ,Mechanical system ,22E70 ,Quantum mechanics ,Quantum dissipation ,Toda lattice ,Quantum statistical mechanics ,Quantum ,Mathematical Physics ,Mathematics - Published
- 1986
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27. Whittaker vectors and conical vectors
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Roe Goodman and Nolan R. Wallach
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Weyl group ,Pure mathematics ,Verma module ,Analytic continuation ,Holomorphic function ,Lie group ,Whittaker–Shannon interpolation formula ,Algebra ,symbols.namesake ,symbols ,Mathematics::Representation Theory ,Whittaker function ,Analysis ,Meromorphic function ,Mathematics - Abstract
A holomorphic family of differential operators of infinite order is constructed that transforms conical vectors for principal series representations of quasi-split, linear, semi-simple Lie groups into Whittaker vectors. Using this transform, it is shown that algebraic Whittaker vectors (as studied by Kostant) extend to ultradistributions of Gevrey type on principal series representations. For each element of the small Weyl group, a meromorphic family of Whittaker vectors is constructed from this transform and the Kunze-Stein intertwining integrals. An explict formula is derived for the smooth Whittaker vector (discovered by Jacquet), in terms of these families of ultradistribution Whittaker vectors. In particular, this gives new proofs of Jacquet's analytic continuation of the smooth Whittaker vector and its functional equation (Jacquet and Schiffman). Applications of the transform are also given to the theory of Verma modules. more...
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- 1980
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28. Holomorphic representations of nilpotent Lie groups
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Roe Goodman
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Algebra ,Pure mathematics ,Representation of a Lie group ,Analytic continuation ,Entire function ,Holomorphic function ,Lie group ,Real form ,(g,K)-module ,Nilpotent group ,Analysis ,Mathematics - Abstract
Let G be a connected, simply-connected complex nilpotent Lie group, and Gr ⊂( G a real form of G. Motivated by the problem of analytic continuation of Banach-space representations of GR to holomorphic representations of G, we construct translation-invariant locally-convex algebras of entire functions on G (generalizing the classical spaces of entire functions of finite exponential order). The dual spaces of these algebras are naturally identified with algebras of left-invariant differential operators of infinite order on G. In connection with analytic continuation of unitary representations of GR, we study the convex cone of entire functions on G whose restrictions to GR are positive-definite, and determine the minimal order of growth at infinity of such functions. more...
- Published
- 1979
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29. Interpolation spaces and unitary representations
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Roe Goodman
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Combinatorics ,Algebra ,Semidirect product ,Unitary representation ,Applied Mathematics ,General Mathematics ,Lie algebra ,Adjoint representation ,Banach space ,Lie group ,Universal enveloping algebra ,(g,K)-module ,Mathematics - Abstract
Let G G be a Lie group, π \pi a unitary representation of G G on a Hilbert space H ( π ) \mathcal {H}(\pi ) , and H k ( π ) {\mathcal {H}^k}(\pi ) the subspace of C k {C^k} vectors for π \pi . By quadratic interpolation there is a continuous scale H s ( π ) {\mathcal {H}^s}(\pi ) , s > 0 s > 0 , of G G -invariant Hilbert spaces. When G = H ⋅ K G = H \cdot K is a semidirect product of closed subgroups, then it is proved that H s ( π ) = H s ( π | H ) ∩ H s ( π | K ) {\mathcal {H}^s}(\pi ) = {\mathcal {H}^s}({\left . \pi \right |_H}) \cap {\mathcal {H}^s}({\left . \pi \right |_K}) for s > 0 s > 0 . For solvable G G this gives a characterisation of H s ( π ) {\mathcal {H}^s}(\pi ) in terms of smoothness along one-parameter subgroups, and an elliptic regularity result. more...
- Published
- 1981
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30. Classical and quantum-mechanical systems of Toda lattice type. I
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Roe Goodman and Nolan R. Wallach
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Generalization ,Mathematical analysis ,Lie group ,Statistical and Nonlinear Physics ,Type (model theory) ,17B35 ,Hamiltonian system ,58F06 ,22E70 ,Factorization ,Affine transformation ,81C99 ,Toda lattice ,Quantum ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
Solutions to the classical periodic and non-periodic Toda lattice type Hamiltonian systems are expressed in terms of an Iwasawa-type factorization of a “large” Lie group. The scattering of these systems is determined in the non-periodic case. For the generalized periodic Toda lattices a generalization of Kostant's formula is obtained using standard representations of affine Lie groups. more...
- Published
- 1982
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31. Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle
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Nolan R. Wallach and Roe Goodman
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Pure mathematics ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Simple Lie group ,(g,K)-module ,Stone–von Neumann theorem ,Algebra ,Loop (topology) ,symbols.namesake ,Representation of a Lie group ,Group of Lie type ,Representation theory of SU ,symbols ,Mathematics - Published
- 1984
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32. Controlled Selection—A Technique in Probability Sampling
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Leslie Kish and Roe Goodman
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Statistics and Probability ,Basis (linear algebra) ,Statistics ,Sampling (statistics) ,Cluster sampling ,Statistics, Probability and Uncertainty ,Selection (genetic algorithm) ,Probability sampling ,Mathematics ,Stratified sampling - Abstract
A sampling technique is defined as introducing control into the selection of n out of N sampling units when it increases the probabilities of selection for preferred combinations of units (and decreases the probabilities for non-preferred combinations). Methods used in the past have by no means exhausted the possibilities of controlled selection, however. Procedures are developed by which the probabilities of selection for preferred combinations are sharply increased and the theoretical basis for the methods is stated. The methods are applied to a specific problem and the procedures are described in detail. It it found that as a result the variances of estimates for several important items are reduced as compared with the corresponding variances for stratified random sampling. * Presented at the 108th Annual Meeting of the American Statistical Association, Cleveland, December 29, 1948. more...
- Published
- 1950
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33. On the boundedness and unboundedness of certain convolution operators on nilpotent Lie groups
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Roe Goodman
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Discrete mathematics ,Nilpotent ,Pure mathematics ,Representation of a Lie group ,Applied Mathematics ,General Mathematics ,Simple Lie group ,Adjoint representation ,Lie group ,Nilpotent group ,Central series ,Haar measure ,Mathematics - Abstract
One method of proving irreducibility of the “principal series” representations of semisimple Lie groups involves showing that a certain nonintegrable function on a nilpotent subgroup X X cannot be regularized to give a bounded convolution operator on L 2 ( X ) {L_2}(X) . This note gives an elementary proof of this unboundedness property for the groups X X which occur in real-rank one semisimple groups. more...
- Published
- 1973
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34. Sampling for the 1947 Survey of Consumer Finances
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Roe Goodman
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Statistics and Probability ,Survey methodology ,Geography ,Statistics ,Econometrics ,Sampling (statistics) ,Survey sampling ,Sample (statistics) ,Cluster sampling ,Lot quality assurance sampling ,Statistics, Probability and Uncertainty ,Sampling frame ,Stratified sampling - Abstract
The designing of a sample for financial surveys requires special attention to families with higher incomes in order that adequate information may be obtained about what the “dollars will do” as well as what the heads of households will do. The chief problem posed here is the adaptation of area sampling to a procedure that is more efficient than the more usual stratified random sampling would be. The modifications introduced were essentially devices for disproportionate sampling of households classified by average rent values and estimated income levels. In determining sample size it must be recognized that the sampling errors of different items of information vary considerably from item to item. The findings from the 1947 Survey of Consumer Finances have been published in the June, July and August issues of the Federal Reserve Bulletin. The methods and procedures illustrate the application of area sampling to a practical survey problem. The sample consisted of small clusters of households within ... more...
- Published
- 1947
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35. On localization and domains of uniqueness
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Roe Goodman
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Pure mathematics ,Applied Mathematics ,General Mathematics ,Uniqueness ,Mathematics - Published
- 1967
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36. Differential Operators of Infinite Order on a Lie Group, II
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Roe Goodman
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Algebra ,Pure mathematics ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Infinite group ,General Mathematics ,Simple Lie group ,Adjoint representation ,Lie group ,Lie theory ,Operator theory ,Mathematics - Published
- 1971
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37. Complex Fourier analysis on a nilpotent Lie group
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Roe Goodman
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Pure mathematics ,Applied Mathematics ,General Mathematics ,Regular representation ,Adjoint representation ,Lie group ,(g,K)-module ,Algebra ,symbols.namesake ,Nilpotent ,Representation of a Lie group ,Fourier transform ,symbols ,Nilpotent group ,Mathematics - Abstract
Let G G be a simply-connected nilpotent Lie group, with complexification G c {G_c} . The functions on G G which are analytic vectors for the left regular representation of G G on L 2 ( G ) {L_2}(G) are determined in this paper, via a dual characterization in terms of their analytic continuation to G c {G_c} , and by properties of their L 2 {L_2} Fourier transforms. The analytic continuation of these functions is shown to be given by the Fourier inversion formula. An explicit construction is given for a dense space of entire vectors for the left regular representation. In the case G = R G = R this furnishes a group-theoretic setting for results of Paley and Wiener concerning functions holomorphic in a strip. more...
- Published
- 1971
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38. Analytic domination by fractional powers of a positive operator
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Roe Goodman
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Indefinite sum ,Finite-rank operator ,Compact operator ,Shift operator ,Differential operator ,Strictly singular operator ,Quasinormal operator ,Semi-elliptic operator ,Elliptic operator ,Operator (computer programming) ,Multiplication operator ,Hypoelliptic operator ,Analysis ,Mathematics - Abstract
was an elliptic operator of order greater than one, while X was a first order operator. Thus condition (i) would seem unnecessarily strong in these circumstances. In the concrete case of a positive self- adjoint elliptic differential operator A on a subset of Rn, Kotake and Narasimhan [5] were able to improve Nelson’s results, and this suggested that perhaps one could obtain analytic domination of X by a fractional power of more...
- Published
- 1969
- Full Text
- View/download PDF
39. One-parameter groups generated by operators in an enveloping algebra
- Author
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Roe Goodman
- Subjects
Algebra ,Unitary representation ,Subalgebra ,Adjoint representation ,Zonal spherical function ,Universal enveloping algebra ,(g,K)-module ,Casimir element ,Analysis ,Mathematics ,Graded Lie algebra - Abstract
Let π be a unitary representation of a connected Lie group G, and let ∂π be the associated representation of the complex enveloping algebra U(G)C of the Lie algebra G. Let h be a commutative subalgebra of G. The commutation relation eTXe−T = eadTX, (∗) with X ϵ ∂π(G) and T a skew-Hermitian element of ∂π(U(h)C), is established as an operator identity on the space of differentiable vectors for π, under the hypothesis that adG(h) is nilpotent. Relation (∗) is then used to prove that a spectral subspace corresponding to a compact, connected component of the spectrum of T is invariant under π(G). more...
- Full Text
- View/download PDF
40. Projective unitary positive-energy representations of diff (S1)
- Author
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Roe Goodman and Nolan R. Wallach
- Subjects
Classical group ,Discrete mathematics ,Unitary representation ,Representation of a Lie group ,Projective unitary group ,Unitary group ,Projective linear group ,Representation theory ,Analysis ,Projective representation ,Mathematics - Abstract
Let D be the group of orientation-preserving diffeomorphisms of the circle S1. Then D is Frechet Lie group with Lie algebra (δ∞)R the smooth real vector fields on S1. Let δR be the subalgebra of real vector fields with finite Fourier series. It is proved that every infinitesimally unitary projective positive-energy representation of δR integrates to a continuous projective unitary representation of D . This result was conjectured by V. Kac. more...
- Full Text
- View/download PDF
41. Elliptic and subelliptic estimates for operators in an enveloping algebra
- Author
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Roe Goodman
- Subjects
Filtered algebra ,Algebra ,General Mathematics ,47G05 ,Universal enveloping algebra ,35J99 ,Algebra over a field ,Mathematics ,17B35 ,22E30 - Published
- 1980
42. Filtrations and asymptotic automorphisms on nilpotent Lie groups
- Author
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Roe Goodman
- Subjects
Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Simple Lie group ,Adjoint representation ,Cartan subalgebra ,Central series ,Graded Lie algebra ,22E60 ,Nilpotent ,Representation of a Lie group ,22E25 ,Geometry and Topology ,Nilpotent group ,Analysis ,Mathematics - Published
- 1977
- Full Text
- View/download PDF
43. Approximating lie algebras of vector fields by nilpotent lie algebras
- Author
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Roe Goodman
- Subjects
Algebra ,Pure mathematics ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Non-associative algebra ,Lie bracket of vector fields ,Killing form ,Affine Lie algebra ,Lie conformal algebra ,Mathematics ,Graded Lie algebra - Published
- 1979
- Full Text
- View/download PDF
44. Positive-Energy Representations of the Group of Diffeomorphisms of the Circle
- Author
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Roe Goodman and Nolan R. Wallach
- Subjects
Pure mathematics ,Unitary representation ,Group (mathematics) ,Subalgebra ,Lie algebra ,Lie group ,Fourier series ,Unitary state ,Mathematics ,Projective representation - Abstract
Let D be the group of orientation-preserving diffeomorphisms of the circle S1. Then D is Frechet Lie group with Lie algebra (d∞)ℝ the smooth real vector fields on S1. Let dℝ be the subalgebra of real vector fields with finite Fourier series. This lecture outlines a proof that every infinitesimally unitary projective positive-energy representation of dℝ integrates to a continuous projective unitary representation of D. This result was conjectured by V. Kac. more...
- Published
- 1985
- Full Text
- View/download PDF
45. Classical and quantum mechanical systems of Toda-lattice type. II. Solutions of the classical flows
- Author
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Roe Goodman and Nolan R. Wallach
- Subjects
Physics ,22E70 ,58F06 ,Statistical and Nonlinear Physics ,82A68 ,Toda lattice ,58F05 ,Quantum ,Mathematical Physics ,Mathematical physics - Abstract
Les solutions des systemes hamiltoniens de type reseau de Toda periodiques et non periodiques classiques sont exprimees en fonction d'une factorisation de type Iwasawa d'un grand groupe de Lie. On determine la diffusion de ces systemes dans le cas non periodique. Pour les reseaux de Toda periodiques generalises, on obtient une generalisation de la formule de Kostant en utilisant des representations standards des groupes de Lie affine more...
- Published
- 1984
46. Nilpotent Lie Groups
- Author
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Roe Goodman
- Subjects
Pure mathematics ,Representation of a Lie group ,Simple Lie group ,Cartan subalgebra ,Lie group ,Real form ,Nilpotent group ,Central series ,Mathematics ,Graded Lie algebra - Published
- 1976
- Full Text
- View/download PDF
47. Singular integral operators on nilpotent Lie groups
- Author
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Roe Goodman
- Subjects
Algebra ,Nilpotent ,Representation of a Lie group ,General Mathematics ,Simple Lie group ,Adjoint representation ,Lie group ,Singular integral ,Nilpotent group ,Central series ,Mathematics - Published
- 1980
48. Filtrations and canonical coordinates on nilpotent Lie groups
- Author
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Roe Goodman
- Subjects
Nilpotent Lie algebra ,Discrete mathematics ,Pure mathematics ,Structure constants ,Applied Mathematics ,General Mathematics ,Simple Lie group ,Lie algebra ,Adjoint representation ,Universal enveloping algebra ,(g,K)-module ,Graded Lie algebra ,Mathematics - Abstract
Let g be a finite-dimensional nilpotent Lie algebra over a field of characteristic zero. Introducing the notion of a positive, decreasing filtration 9 on a, the paper studies the multiplicative structure of the universal enveloping algebra {/(g), and also transformation laws between ^-canonical coordinates of the first and second kind associated with the Campbell- Hausdorff group structure on g. The basic technique is to exploit the duality between C/(g) and S(g*), the symmetric algebra of g*, making use of the filtration S". When the field is the complex numbers, the preceding results, together with the Cauchy estimates, are used to obtain estimates for the structure constants for l/(g). These estimates are applied to construct a family of completions 1/(0)98, °i U(S)> on which the corresponding simply- connected Lie group G acts by an extension of the adjoint representation. more...
- Published
- 1978
- Full Text
- View/download PDF
49. Questions and Answers
- Author
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Frederick Mosteller, Roe Goodman, and E. J. Gumbel
- Subjects
Statistics and Probability ,General Mathematics ,Statistics, Probability and Uncertainty - Published
- 1948
- Full Text
- View/download PDF
50. Analytic and entire vectors for representations of Lie groups
- Author
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Roe Goodman
- Subjects
Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Representation theory of SU ,Applied Mathematics ,General Mathematics ,Simple Lie group ,Fundamental representation ,Lie group ,Lie theory ,Representation theory ,Mathematics - Published
- 1969
- Full Text
- View/download PDF
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