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We construct a monoidal category C w , v which categorifies the doubly-invariant algebra C N ′ ( w ) [ N ] N ( v ) associated with Weyl group elements w and v. It gives, after a localization, the coordinate algebra C [ R w , v ] of the open Richardson variety associated with w and v. The category C w , v is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v = id , C w , v is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra A q ( n ( w ) ) Z [ q , q − 1 ] given by Kang–Kashiwara–Kim–Oh. We show that the category C w , v contains special determinantial modules M ( w ≤ k Λ , v ≤ k Λ ) for k = 1 , … , l ( w ) , which commute with each other. When the quiver Hecke algebra R is symmetric, we find a formula of the degree of R-matrices between the determinantial modules M ( w ≤ k Λ , v ≤ k Λ ) . When it is of finite ADE type, we further prove that there is an equivalence of categories between C w , v and C u for w , u , v ∈ W with w = v u and l ( w ) = l ( v ) + l ( u ) .

We investigate support schemes for infinitesimal unipotent supergroups and their representations. Our main results provide a non-cohomological description of these schemes which generalizes the classical work of Suslin, Friedlander, and Bendel. As a consequence, support schemes in this setting have the desired features of such a theory, including naturality with respect to group homomorphisms, the tensor product property, and realizability. As an application of the theory developed here, we investigate support varieties for certain finite-dimensional Hopf subalgebras of the Steenrod algebra., Comment: 45 pages. Comments welcome

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry., Comment: 25 pages

For all primes p and for all fields, we find a sufficient and necessary condition of the existence of a unipotent Galois extension of degree p 6 . The main goal of this paper is to describe an explicit construction of such a Galois extension over fields admitting such a Galois extension. This construction is surprising in its simplicity and generality. The problem of finding such a construction has been left open since 2003. Recently a possible solution of this problem gained urgency because of an effort to extend new advances in Galois theory and its relations with Massey products in Galois cohomology.

We introduce the notion of spectral transfer morphisms between normalized affine Hecke algebras, and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine Hecke algebras involved. We define a partial ordering on the set of isomorphism classes of normalized affine Hecke algebras, which plays an important role for the Langlands parameters of Lusztig's unipotent representations., Comment: 38 pages; The ordering of the material has been improved in this version

This is a survey article in which we discuss some recent approach toward the classification of global factorizable integrals, and constructions of Langlands functorial lifting using small representations. To do that we use the language of unipotent orbits and their relation to Fourier coefficients. We also introduce the dimension equation, and set our goal to classify all global integrals of a certain form which satisfies this equation.

We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms. A reductive symmetric space is a homogeneous space G / H for a reductive Lie group G of the Harish-Chandra class, where H is an open subgroup of the fixed-point subgroup for an involution σ on G . Let P be a parabolic subgroup such that σ ( P ) is opposite to P and let N P be the unipotent radical of P . For a compactly supported smooth function ϕ on G / H , we define R P ( ϕ ) ( g ) to be the integral of N P ∋ n ↦ ϕ ( g n ⋅ H ) over N P . The Radon transform R P thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions. For these transforms we derive support theorems in which the support of ϕ is (partially) characterized in terms of the support of R P ϕ . The proof is based on the relation between the Radon transform and the Fourier transform on G / H , and a Paley–Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.

We give a combinatorial description of the Springer correspondence for classical Lie algebras $\mathfrak{g}$ of type $B,C$ or $D$ and their duals $\mathfrak{g}^*$ in characteristic 2. The combinatorics used here is of the same kind as those appearing in the description of (generalized) Springer correspondence for unipotent case of classical groups $G$ by Lusztig in odd characteristic and by Lusztig and Spaltentstein in characteristic 2. It is very nice that this combinatorics gives a unified description for (generalized) Springer correspondences of classical groups in all cases, namely, in $G$, $\mathfrak{g}$ and $\mathfrak{g}^*$ in all characteristics. Moreover, it gives rise to close formulas for computing the correspondences., Comment: 30 pages

We construct a family of orthogonal characters of an algebra group which decompose the supercharacters defined by Diaconis and Isaacs. Like supercharacters, these characters are given by nonnegative integer linear combinations of Kirillov functions and are induced from linear supercharacters of certain algebra subgroups. We derive a formula for these characters and give a condition for their irreducibility; generalizing a theorem of Otto, we also show that each such character has the same number of Kirillov functions and irreducible characters as constituents. In proving these results, we observe as an application how a recent computation by Evseev implies that every irreducible character of the unitriangular group $\UT_n(q)$ of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements is a Kirillov function if and only if $n\leq 12$. As a further application, we discuss some more general conditions showing that Kirillov functions are characters, and describe some results related to counting the irreducible constituents of supercharacters., 22 pages

The standard supercharacter theory of the finite unipotent upper-triangular matrices U n ( q ) gives rise to beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of U m ( q ) ⊆ U n ( q ) for m ⩽ n lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of U n ( q ) is a nonnegative integer linear combination of supercharacters of U m ( q ) (in fact, it is polynomial in q). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of U n ( q ) , this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs.

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construction (called "S2-extension") in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical "S2-ification" of appropriate X. The construction also has applications to the "Macaulayfication" problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown in the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S2-extension to give a uniform construction of the desired variety., Comment: 30 pages; minor corrections and additions

This paper deals with graded representations of the symmetric group on the cohomology ring of flags fixed by a unipotent matrix. We consider a combinatorial property, called the “coincidence of dimension” of the graded representations, and give an interpretation in terms of representation theory of the symmetric group in the case where the corresponding partition of the unipotent matrix is a hook or a rectangle. The interpretation is equivalent to a recursive formula of Green polynomials at roots of unity.

This paper presents a combinatorial approach to the tempered representation theory of a graded Hecke algebra H of classical type B or C, with arbitrary parameters. We present various combinatorial results which together give a uniform combinatorial description of what becomes the Springer correspondence in the classical situation of equal parameters. More precisely, by using a general version of Lusztig's symbols which describe the classical Springer correspondence, we associate to a discrete series representation of H with central character W0c, a set Σ(W0c) of W0-characters (where W0 is the Weyl group). This set Σ(W0c) is shown to parametrize the central characters of the generic algebra which specialize into W0c. Using the parabolic classification of the central characters of H^t(R) on the one hand, and a truncated induction of Weyl group characters on the other hand, we define a set Σ(W0c) for any central character W0c of H^t(R), and show that this property is preserved. We show that in the equal parameter situation we retrieve the classical Springer correspondence, by considering a set U of partitions which replaces the unipotent classes of SO2n+1(C) and Sp2n(C), and a bijection between U and the central characters of H^t(R). We end with a conjecture, which basically states that our generalized Springer correspondence determines H^t(R) exactly as the classical Springer correspondence does in the equal label case. In particular, we conjecture that Σ(W0c) indexes the modules in H^t(R) with central character W0c, in the following way. A module M in H^t(R) has a natural grading for the action of W0, and the W0-representation χ(M) in its top degree is irreducible. When M runs through the modules in H^t(R) with central character W0c, χ(M) runs through Σ(W0c). Moreover, still in analogy with the equal parameter case, we conjecture that the W0-structure of the modules in H^t(R) can be computed using (generalized) Green functions.

For a reductive, algebraic group, G, the Steinberg variety of G is the set of all triples consisting of a unipotent element, u, in G and two Borel subgroups of G that contain u. We define generalized Steinberg varieties that depend on four parameters and analyze in detail two special cases that turn out to be related to distinguished double coset representatives in the Weyl group. Using one of the two special cases, we define a parabolic version of a map from the Weyl group to a set of nilpotent orbits of G in Lie(G) defined by Joseph and study some of its properties.

We establish a general multiple recurrence theorem for an action of a nilpotent group by homeomorphisms of a compact space. This theorem can be viewed as a nilpotent version of our recent polynomial Hales–Jewett theorem (Ann. Math. 150 (1999) 33) and contains nilpotent extensions of many known “abelian” results as special cases.

Let U ( G ) be a maximal unipotent subgroup of one of the classical groups G = GL ( V ), O ( V ), Sp ( V ). Let W be a direct sum of copies of V and its dual V *. For the natural action U ( G ) : W , we describe a minimal system of homogeneous generators for the algebra of U ( G )-invariant regular functions on W . For G = O ( V ), Sp ( V ), this result is connected with a construction for the irreducible representations of G due to H. Weyl.

Let G be a connected reductive complex algebraic group with Lie algebra Lie G=g. Let B be a Borel subgroup of G with unipotent radical Bu . We denote the Lie algebra of B and Bu by b and bu , respectively. The group B acts on any ideal of b by means of the adjoint representation. After a preliminary section we study a relationship between spherical nilpotent orbits and abelian ideals a of b, using the structure theory for these orbits from [10]. The principal result of this section is that, for an abelian ideal a of b, any nilpotent orbit meeting a is a spherical G-variety, see Theorem 2.3. As a consequence of this we obtain a short conceptual proof of a finiteness theorem from [14]. Namely, for a parabolic subgroup P of G and an abelian ideal a of p=Lie P in the nilpotent radical pu=Lie Pu , the group P operates on a with finitely many orbits. The proof of this fact in [14] involved long and tedious case by case considerations. We also prove a partial converse to the result just mentioned. Following [4], we say that an ideal of b is ad nilpotent whenever it consists of nilpotent elements. In case G is simply laced, we show that an ad nilpotent ideal c of b is abelian provided any nilpotent orbit meeting c is spherical, see Proposition 2.7. In Section 3 we consider some properties of ad nilpotent ideals of b. In Theorem 3.2 we give a description of the normaliser of such ideals. This applies in particular to abelian ideals of b. A remarkable theorem of doi:10.1006 aima.2000.1959, available online at http: www.idealibrary.com on

In this paper we prove the results announced in [13]. Let G be a semi- simple, simply connected algebraic group defined over an algebraically closed field k. Let T be a maximal torus, B a Bore1 subgroup, B 3 T. Let W be the Weyl group of G. Let R (resp. R+ ) be the set of roots (resp. positive roots) relative to T (resp. B). Let S be the set of simple roots in R+. Let P be a maximal parabolic subgroup in G with associated fundamental weight w. Let W, be the Weyl group of P, and Wp be the set of minimal representatives of W/W,. For w E Wp, let e(w) be the point and X(w) the Schubert variety in G/P associated to w. In this paper we deter- mine the multiplicity m,(w) of X(w) at e(z), where e(z) E X(w), for all minuscule P’s and also for P = Pgn, G being of type C, (here Pun denotes the maximal parabolic subgroup obtained by omitting a,). The determina- tion of m,(w) is done as follows. Let L be the ample generator of Pic(G/P). A basis has been constructed for @(X(w), L”) in terms of standard monomials on X(w) (cf. [ 16, 11 I). Let U; be the unipotent subgroup of G generated by U-,, /?ET(R+ - Rp+) (here R, denotes the set of roots of P and U, denotes the unipotent subgroup of G, associated to tl E R). Then U; e(r) gives an affme neighborhood of e(z) in G/P. Let A, be the affine algebra of U, e(z) and A,.. = A,/&, where & is the ideal of elements of A, that vanish on X(w) n U; e(z). Let M,,, be the maximal ideal in A, H, corresponding to e(r). Then using the results of [ 16, 111, we obtain a basis of M;, &f:,+,,’ . This enables us to obtain an inductive formula for F,,,, the Hilbert polynomial of X(w) at e(T) (cf. Corollaries 3.8 and 4.11), and also express m, (w) in terms of m, (w’)‘s, X(w’)‘s being the Schubert divisors in X(w) such that e(T)E X(w’) (cf. Theorems 3.7 and 4.10). Using this we

The unitary highest weight modules of a semisimple Lie group G have been classified [S, 93. Unfortunately this classification does not fit well with a more general picture of the unitary dual of G. We give such a classification for G a classical group, using derived functors and the philosophy of unipotent representations. In particular these representations are associated to coadjoint orbits of G. Let 8, = Lie(G), 0 a Cartan involution of 6 = 8, @C. A derived functor module is defined by the data q = I + u a O-stable parabolic subalgebra of 0, and 17 an irreducible unitary representation of L, the stabilizer of L in G. Considering Z7 as a representation of I, write Z7= (A, rc) where A E c* (c = center of I, * denotes dual), and rt = Z7l r,,,, . We define A(& rc) = &(,I, rc) (cf. Definition 6.1). Now fix n: and vary 1. Under certain restrictions on I, which amount to a positivity condition on the corresponding bundle over G/L, A(I, rr) is known to be irreducible and unitary. As II varies within this range the Langlands parameters of A(A, rr) vary smoothly. However, one cannot obtain all unitary highest weight modules under these restrictions. The orbit method suggests how to obtain the remaining unitary highest weight modules. The modules A(& rr) are associated to coadjoint orbits as follows. If rc is trivial we say A(]“, rr) is associated to the elliptic orbit 0 = G. II. Not all unitary highest weight modules may be obtained with rc trivial, an example of which is the oscillator representation of $(2n, R) (the metapletic group). We define 2n irreducible unitary highest weight modules rc,+ (1 (2(n-k),R)) is 113 OOOl-8708/87 $7.50

In Math. Z. (176 (1981), 359–374) I explicitly determined the invariants of a certain class of unipotent group actions, and obtained a positive partial answer to Hilbert's 14th problem for nonreductive groups. The class of groups for which the method worked remained quite obscure. Theorem (4.2) of the present paper gives a precise description of the cases where the algebras of invariants are spanned by standard bitableaux, hence have a straightening law. The unipotent groups in question (“radizielle Untergruppen” of GLn) correspond, up to conjugation, to finite (partially) ordered sets. The promised description is done by properties of the ordered sets that are easy to test. This is another example where combinatorial methods are important for the theory of invariants.

We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. In particular, there are no dense double cosets of the centralizer of a noncentral element. This result has been used by Prasad in considering Tits systems for psuedoreductive groups. Our final result is a generalization of the Baer-Suzuki theorem for p-elements with p a prime at least 5., Comment: 36 pages

Let G be an algebraic reductive group over a field of positive characteristic. Choose a parabolic subgroup P in G and denote by U its unipotent radical. Let X be a G -variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of l-adic sheaves on X with respect to a generic character χ:U→ G a commutes with Verdier duality. Namely, in the first example we take X to be an arbitrary G -variety and we prove the above property for all U -equivariant sheaves on X where U is the unipotent radical of an opposite parabolic subgroup; in the second example we take X = G and we prove the corresponding result for sheaves which are equivariant under the adjoint action (the latter result was conjectured by B. C. Ngo who proved it for G = GL ( n )). As an application of the proof of the first statement we reprove a theorem of N. Katz and G. Laumon about local acyclicity of the kernel of the Fourier–Deligne transform.

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras., Comment: To Appear in Advances in Mathematics (2012), 23 pages

Suppose that W is a Weyl group, let C ( W ) be a space of functions on W , with complex values, invariant under conjugation. We can define an “elliptic scalar product” on C ( W ) . It is a natural ingredient to the representation theory of p -adic reductive groups. Let G be a reductive group over the algebraic closure of a finite field. The generalized Springer correspondence gives a bijection between two sets: – the set of pairs ( U , E ) , where U is an unipotent orbit of G and E is a G -equivariant irreducible local system on U ; – the disjoint union of the sets of irreducible representations of certain Weyl groups related to G . Using Kazhdan–Lusztig polynomials, we modify the generalized Springer correspondence. By the modified correspondence, a pair ( U , E ) as above maps to a representation of a certain Weyl group, and this representation is, in general, reducible. There is no simple formula that relates the elliptic scalar product and the generalized Springer correspondence. But a simple formula does exist, and we prove it, that relates the elliptic scalar product and the modified generalized Springer correspondence. Our result is, in fact, a corollary of a theorem of Lusztig on the restriction of character-sheaves to the unipotent variety.

Let u be a unipotent element in a complex semisimple group G and let 3,, be the variety of Bore1 subgroups of G which contain u. In [4, 51, Springer has defined a representation of W, the Weyl group of G, on the homology of 2:, and showed how to decompose the Wrepresentation in the top homology of .53’U so that all irreducible representations of W are obtained. His mehod used etale cohomology of algebraic varieties in characteristic p. In this paper, we shall give an elementary construction (independent of a he conjectured that the representation in the top homology of 2 is the two-sided regular representation of W and showed how this could be used to prove the completeness of the set of W-representation in the top homologies of z#,,.

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring K[X]G in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes K[X]G in terms of a so-called colon-operation. From this, generators of K[X]G can be obtained in finite time if it is finitely generated. Under the additional hypothesis that K[X] is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is K[X]G. Along the way, we develop some techniques for dealing with nonfinitely generated algebras. In particular, we introduce the finite generation ideal.

1.1 Let G(F,) be the group of Fq rational points of a connected reductive algebraic group G defined over a finite field F,. Throughout this paper we shall assume that the characteristic of the ground field is large. The following is part of Problem II in [L9]. “Let p be an irreducible complex character of G(F,). Show that there is a unique umpotent class c’ in G which has the property that Cga c’CFqI p(g) # 0 and has maximal dimension among classes with this property.” One of the results of this paper is a solution of this problem; we also prove the following refinement. If g E GF is such that Tr(g, p) # 0 then the unipotent part of g lies in C or in a conjugacy class of dimension