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Isotropic odd unitary groups generalize Chevalley groups of classical types over commutative rings and their twisted forms. Such groups have root subgroups parameterized by a root system $\mathsf{BC}_\ell$ and may be constructed by so-called odd form rings with Peirce decompositions. We show the converse: if a group $G$ has root subgroups indexed by roots of $\mathsf{BC}_\ell$ and satisfying natural conditions, then there is a homomorphism $\mathrm{StU}(R, \Delta) \to G$ inducing isomorphisms on the root subgroups, where $\mathrm{StU}(R, \Delta)$ is the odd unitary Steinberg group constructed by an odd form ring $(R, \Delta)$ with a Peirce decomposition. For groups with root subgroups indexed by $\mathsf A_\ell$ (the already known case) the resulting odd form ring is essentially a generalized matrix ring., Comment: minor corrections

Using cohomological methods, we show that lattices in semisimple groups are typically stable with respect to the Frobenius norm but not with respect to the operator norm., Comment: incorporated remarks and corrections of the reviewers

If $A$ is a unital associative ring and $\ell \geq 2$, then the general linear group $\mathrm{GL}(\ell, A)$ has root subgroups $U_\alpha$ and Weyl elements $n_\alpha$ for $\alpha$ from the root system of type $\mathsf A_{\ell - 1}$. Conversely, if an arbitrary group has such root subgroups and Weyl elements for $\ell \geq 4$ satisfying natural conditions, then there is a way to recover the ring $A$. We prove a generalization of this result not using the Weyl elements, so instead of the matrix ring $\mathrm M(\ell, A)$ we construct a non-unital associative ring with a well-behaved Peirce decomposition., Comment: Some misprints are fixed

We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusztig, and that the Kazhdan-Lusztig map is a section of our map. This settles several conjectures in the literature. For classical groups, we prove more refined results by introducing and studying the "skeleta" of affine Springer fibers., Comment: minor changes from the published version

We give an upper and lower estimate for the maximal commutator length of a noncentral element of the elementary subgroup of the general linear group over a skew-field based on the maximal commutator length of an element of the multiplicative group of that skew-field., Comment: 12 pages, to be published in Journal of Mathematical Sciences

Let $U_1, U_2$ be connected commutative unipotent algebraic groups defined over an algebraically closed field $k$ of characteristic $p>0$ and let $\mathcal{L}$ be a bimultiplicative $\overline{\mathbb{Q}}_\ell$-local system on $U_1\times U_2$. In this paper we will study the $\overline{\mathbb{Q}}_\ell$-cohomology $H^*_c(U_1\times U_2,\mathcal{L})$, which turns out to be supported in only one degree. We will construct a finite Heisenberg group $\Gamma$ which naturally acts on $H^*_c(U_1\times U_2,\mathcal{L})$ as an irreducible representation. We will give two explicit realizations of this cohomology and describe the relationship between these two realizations as a finite Fourier transform., Comment: 19 pages

In this paper we prove that the affine Schur algebra $\s$ is affine cellular over $\bbq$. As an application, we prove it is of finite global dimension., Comment: 13 pages

By using perverse sheaves on representation spaces of quivers over $k[t]/(t^n)$ and jet schemes over flag varieties, we construct a geometric composition algebra $\mathbf K$ under Lusztig's framework on geometric realizations of the negative part of quantum algebras. Simple perverse sheaves in $\mathbf K$ form the canonical basis of $\mathbf K$. The relationships among the algebra $\mathbf K$, the composition algebra of locally projective representations of quivers over $k[t]/(t^n)$ and quantum generalized Kac-Moody algebra are provided., Comment: Simplify some proofs and some other revisions on the earlier version

We show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the Weyl group., Comment: 31 pages

We compute the integral cohomology of the minimal non-trivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the sub-root system generated by the long simple roots. The modulo $\ell$ reduction of the Springer correspondent representation involves the sign representation exactly when $\ell$ divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes., Comment: 29 pages, v2 : Leray-Serre spectral sequence replaced by Gysin sequence only, corrected typos

Let G be a special orthogonal group over an algebraically closed field of characteristic exponent p. In this paper we extend certain aspects of the Dynkin-Kostant theory of unipotent elements in G (when p=1) to the general case (including p=2)., Comment: 25 pages

We study a class of perverse sheaves on some spherical varieties which include the strata of the De Concini-Procesi completion of a symmetric variety. This is a generalization of the theory of (parabolic) character sheaves., Comment: 38 pages

Let $G$ be a semisimple adjoint group over $\bold C$ and $\bar{G}$ be the De Concini-Procesi completion of $G$. In this paper, we define a Lagrangian subvariety $\Lambda$ of the cotangent bundle of $\bar{G}$ such that the singular support of any character sheaf on $\bar{G}$ is contained in $\Lambda$., Comment: 8 pages

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in \cite{L3}. We also prove some properties of the character sheaves on the group compactification., Comment: 22 pages. Final version

Let (W, S) be a Coxeter system. We investigate combinatorially certain partial orders, called extended Bruhat orders, on a (W x W)-set W(N,C), which depends on W, a subset N of S, and a component C of N. We determine the length of the maximal chains between two elements. These posets generalize W equipped with its Bruhat order. They include the (W x W)-orbits of the Renner monoids of reductive algebraic monoids and of some infinite dimensional generalizations which are equipped with the partial orders obtained by the closure relations of the Bruhat and Birkhoff cells. They also include the (W x W)-orbits of certain posets obtained by generalizing the closure relation of the Bruhat cells of the wonderful compactification., Comment: 21 pages

This paper contains an exposition of the theory of character sheaves for reductive groups and some attempts to extend it to other cases: unipotent groups, reductive groups modulo the unipotent radical of a parabolic., Comment: 12 pages

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The "trichotomy" here is a case division of the minimal counterexamples within odd type, i.e. groups with a divisibble connected component of the Sylow 2-subgroup. We introduce a charateristic zero notion of unipotence which can be used to obtain a connected nilpotent signalizer functor from any sufficiently non-trivial solvable signalizer functor. This result plugs seamlessly into Borovik's work to eliminate the assumption of tameness from his trichotomy theorem for odd type groups. This work also provides us with a form of Borovik's theorem for degenerate type groups.

We begin the study of character sheaves on a not necessarily connected reductive group, extending the known theory for connected groups., Comment: 33 pages

Let G(K) be the group of K-rational points of a connected adjoint simple algebraic group defined over a non-archimedean local field K. In this paper we classify the unipotent representations of G(K) in terms of the geometry of the Langlands dual group. (This was known earlier in the special case where G(K) is an inner form of a split group.) We also determine which representations are tempered or square integrable., Comment: 52 pages

In this paper we study the Hecke algebra associated with a complex reflection group W. We discuss some properties of the Galois group of the splitting field of this algebra, and study its action on the so-called fake degrees of W. The methods we use to study the Hecke algebra are based on the construction of representations of this algebra as monodromy of equations of Knizhnik-Zamolodchikov type., Comment: 27 pages, no figures. These notes were prepared by K Taniguchi on the bases of two lectures that were presented by the author at RIMS, Kyoto university (Japan) in the fall of 1997. The files are in latex format