Unipotent representations of complex groups and extended Sommers duality

A linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and $\dim G=\dim V$ . A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.

This book is an expanded version of the Hermann Weyl Lectures given at the Institute for Advanced Study in January 1986. It outlines some of what is now known about irreducible unitary representations of real reductive groups, providing fairly complete definitions and references, and sketches (at least) of most proofs. The first half of the book is devoted to the three more or less understood constructions of such representations: parabolic induction, complementary series, and cohomological parabolic induction. This culminates in the description of all irreducible unitary representation of the general linear groups. For other groups, one expects to need a new construction, giving "unipotent representations." The latter half of the book explains the evidence for that expectation and suggests a partial definition of unipotent representations.

Let G be a compact Lie group or a complex reductive affine algebraic group. We explore induced mappings between G-character varieties of surface groups by mappings between corresponding surfaces. It is shown that these mappings are generally Poisson. We also given an effective algorithm to compute the Poisson bi-vectors when G=SL(2,C). We demonstrate this algorithm by explicitly calculating the Poisson bi-vector for the 5-holed sphere, the first example for an Euler characteristic -3 surface.

This paper establishes a functorial algebraic isomorphism between the moduli space BCps(Σ,G) of polystable principal G-bundles with prescribed monodromy on a punctured Riemann surface Σ of genus g≥2, for a complex reductive Lie group G, and the character variety MCK(Σ*,G) of representations of its fundamental group with relatively compact image. The dimension formula dimBCps(Σ,G)=2(g−1)dimC(G)+∑i=1kdimR(Ci), where C1,…,Ck are conjugacy classes in a maximal compact subgroup K⊂G, is derived for complex reductive Lie groups, and singularities are characterized as polystable bundles with non-trivial automorphism groups. As applications of the above geometric results to control theory, it is proved that topologically distinct polystable robotic navigation strategies around obstacles are classified by this character variety. The geometry of singular points in families of polystable control strategies is further investigated, revealing enhanced stability properties characterized by reduced tangent space dimensions arising from non-trivial automorphism groups.

Let $G$ be a complex reductive algebraic group, $g$ its Lie algebra and $h$ a reductive subalgebra of $g$, $n$ a positive integer. Consider the diagonal actions $G:g^n, N_G(h):h^n$. We study a relation between the algebra $C[h^n]^{N_G(h)}$ and its subalgebra consisting of restrictions to $h^n$ of elements of $C[g^n]^G$.

We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cell representations of the Weyl group in the same two-sided cell.We use this map to parametrize the unipotent representations.0.1.Let G be a simple algebraic group defined and split over a finite field F q .Let U be the set of isomorphism classes of irreducible unipotent representations (over C) of the finite group G(F q ).Let W be the Weyl group of G and let Irr(W ) be the set of isomorphism classes of irreducible representations (over C) of W .In [2] a partition of Irr(W ) into families is described and in [4] a partition U = ⊔ c U c of U (with c running over the families of Irr(W )) is introduced.Moreover, in [4, §4] to any family c we have associated a finite group G c and a bijection (a) U c ↔ M (G c ).Here, for any finite group Γ, M (Γ) is the set of Γ-conjugacy classes of pairs (x, ρ) were x ∈ Γ and ρ is an irreducible representation (over C) of the centralizer Z Γ (x) of x in Γ; let C[M (Γ)] (resp.N[M (Γ)]) be the vector space of formal C-linear combinations of elements in M (Γ) and let A Γ : C[M (Γ)] → C[M (Γ)] be the non-abelian Fourier transform of [2] (a linear isomorphism with square 1).Let N[M (Γ)] (resp.R ≥0 [M (Γ)]) be the set of vectors of C[M (Γ)] which are linear combinations with coefficients in N

The asymptotic root distribution is computed for systems of matrix functions associated with finite-dimensional holomorphic representations of a Lie group. This distribution can be expressed via the increments of the representations involved. If the group is reductive, then the number of equations in the system can be arbitrary, from 1 to the dimension of the group. In this case, the computation results are stated in the language of convex geometry. These computations imply the previously known formulas for the density of the solution variety of a system of exponential equations as well as for the number of solutions of a polynomial system and, more generally, of a system formed by matrix functions of representations of a complex reductive Lie group.

This dissertation is devoted to the study of quantizations of nilpotent orbits and their covers, and their applications to the theory of unipotent representations. It combines the results from the papers written by the author during his PhD. In the first part, we construct a bijection between the sets of quantizations of a nilpotent orbit in a classical simple Lie algebra, and quantizations of its function algebra. In the second part, we study the geometry of nilpotent orbit covers, and the sets of quantizations of their function algebras. After that, we use the canonical quantizations of these function algebras to give new definitions of unipotent ideals and unipotent representations of a semisimple complex Lie algebra. Our definitions are vindicated by the many favorable properties which these ideals and bimodules enjoy. First of all, we show that both unipotent ideals and bimodules have nice geometric classifications. Unipotent ideals are classified by certain geometrically-defined equivalence classes of covers of nilpotent orbits. Unipotent bimodules are classified by irreducible representations of certain finite groups. We show that all unipotent ideals are maximal, and for classical groups all unipotent bimodules are unitary. In addition, we show that all unipotent bimodules are, as $G$-representations, of a very special form, proving a conjecture of Vogan. Finally, we show that all special unipotent bimodules are unipotent. The final assertion is proved using a certain refinement of Barbasch-Vogan-Lusztig-Spaltenstein duality, inspired by the symplectic duality. Along the way, we establish combinatorial algorithms (in classical and exceptional types) for computing the infinitesimal characters of unipotent ideals--Author's abstract

We determine the unitary dual of the geometric graded Hecke algebras with unequal parameters which appear in Lusztig’s classification of unipotent representations for exceptional p p -adic groups. The largest such algebra is of type F 4 . F_4. Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent representations with real central character of the p p -adic groups. In order for this correspondence to be applicable here, we show (following Lusztig’s geometric classification, and Barbasch and Moy’s original argument) that the set of tempered modules with real central character for a geometric graded Hecke algebra is linearly independent when restricted to the Weyl group.

It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

Let $\mathrm{B}_n$ denote the set of complex square matrices of order $n$ whose Euclidean operator norms are less than one. Its Shilov boundary is the set $\operatorname{U}(n)$ of all unitary matrices. A holomorphic map $\mathrm{B}_m\to\mathrm{B}_n$ is inner if it sends $\operatorname{U}(m)$ to $\operatorname{U}(n)$. On the other hand we consider the group $\operatorname{U}(n+mj)$ and its subgroup $\operatorname{U}(j)$ that is embedded in $\operatorname{U}(n+mj)$ in the block-diagonal way ($m$ blocks $\operatorname{U}(j)$ and a unit block of size $n$). To any conjugacy class of $\operatorname{U}(n+mj)$ with respect to $\operatorname{U}(j)$ we assign a ‘characteristic function’, which is a rational inner map $\mathrm{B}_m\to\mathrm{B}_n$. We show that the class of inner functions that can be obtained as ‘characteristic functions’ is closed with respect to such natural operations as pointwise direct sums, pointwise products, compositions, substitutions into finite-dimensional representations of general linear groups and so on. We also describe explicitly the corresponding operations on conjugacy classes. Bibliography: 24 titles.

We show that if H is a reductive group, then nth roots of conjugacy classes are a finite union of conjugacy classes, and that if G is an algebraic overgroup of H, then the intersection of H with a conjugacy class of G is a finite union of H-conjugacy classes. These results follow from results on finiteness of unipotent classes in an almost simple algebraic group.

In this paper we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps. In particular, a few classical results of Steinberg and Deligne & Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.

Let $G$ be a complex algebraic group and $L$ an algebraic subgroup of $G$. The quotient space $G/L$ is called weakly commutative if a generic orbit of the action $G:T^*(G/L)$ is a coisotropic submanifold. We classify weakly commutative homogeneous spaces $N\leftthreetimes L/L$ in the case where $L$ is a reductive group and the natural representation $L:\mathfrak n/[\mathfrak n,\mathfrak n]$, where $\mathfrak n$ is the tangent algebra of the group $N$, is irreducible.

Let Γ be a connected reductive group over an algebraically closed field F of positive characteristic and let G be the set of fixed points of Γ under some Frobenius map of Γ. Then G is a finite group of Lie type. The p-modular representations of G for p = char F (the “describing” characteristic case) are closely related to rational representations of Γ, and thus results from the theory of reductive algebraic groups can be used to develop the representation theory of finite groups of Lie type in the describing characteristic case. One of the outstanding open questions in this area is to determine the dimensions of the irreducible representations and the decomposition matrix whose entries record multiplicities of irreducible modules as composition factors of Weyl modules. One way to define Weyl modules is by reducing modulo p (p = char F) the irreducible modules of the complex algebraic group of the same type. Here reduction modulo p happens in both, in the field underlying the group and in the field over which the representing matrices are defined, simultaneously. One of the main tools to study these questions is derived from the Weyl group of Γ. If, for example, Γ is a general linear group, the decomposition matrix of the Weyl modules can be calculated from decomposition numbers of Schur algebras, which in turn can be calculated in terms of symmetric groups. We shall discuss this phenomenon in more detail below.