Discrete Series Characters Research Articles

Branes in JT (super)gravity from group theory

In this work, we revisit the end-of-the-world (EOW) brane amplitudes in JT gravity from a BF gauge theoretic perspective. Observing and identifying the correct group theoretic ingredient for a closed EOW brane as a discrete series character, we use the group theory framework as a guide towards formulating the analogous supersymmetric problem. We compute these amplitudes explicitly in the supersymmetric generalizations of JT gravity (N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1, 2, 4), motivated by the prospective of possibly finite amplitudes. In the process, we develop some of the representation theory of OSp(2|2, ℝ) and PSU(1, 1|2), relevant for the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 and N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 cases.

A generalization of combinatorial identities for stable discrete series constants

This article is concerned with the constants that appear in Harish-Chandra’s character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra’s work the only information we have about these constants is that they are uniquely determined by an inductive property. Later, Goresky–Kottwitz–MacPherson (1997) and Herb (2000) gave different formulas for these constants. In this article, we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying the discrete series character identities of Morel (2011). We deduce this identity from a general abstract theorem giving a way to calculate the alternating sum of the values of a valuation on the chambers of a Coxeter arrangement. We also introduce a ring structure on the set of valuations on polyhedral cones in Euclidean space with values in a fixed ring. This gives a theoretical framework for the valuation appearing in Goresky– Kottwitz–MacPherson’s 1997 paper. In an appendix, we extend Herb’s notion of 2-structures to pseudo-root systems.

On a uniqueness property of supercuspidal unipotent representations

The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig [17]), is a rational function of q evaluated at q=q, the cardinality of the residue field. The irreducible factors of this rational function are q and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of M. Reeder [28], and for the remaining exceptional cases this is verified in [7]. In the present paper we treat the classical families.The main result of this article characterizes unramified Lusztig-Langlands parameters which support a cuspidal local system in terms of formal degrees. The result implies the uniqueness of so-called cuspidal spectral transfer morphisms (as introduced in [22]) between unipotent affine Hecke algebras (up to twisting by unramified characters). In [23] the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case.

The Infinitesimal Characters of Discrete Series for Real Spherical Spaces

Let $$Z=G/H$$ be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of G on $$L^2(Z)$$ . It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of $$L^2(Z)$$ , have infinitesimal characters which are real and belong to a lattice. Moreover, let K be a maximal compact subgroup of G. Then each irreducible representation of K occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the discrete components of the space of square integrable sections of a line bundle, given by a unitary character on an abelian extension of H.

Orbital integrals and K-theory classes

Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C^*_rG)\to \mathbb{C}$ defined by orbital integrals to recover group theoretic information about $G$, including information contained in $K$-theory classes not associated to the discrete series. An important tool is a fixed point formula for equivariant indices obtained by the authors in an earlier paper. Applications include a tool to distinguish classes in $K_0(C^*_rG)$, the (known) injectivity of Dirac induction, versions of Selberg's principle in $K$-theory and for matrix coefficients of the discrete series, a Tannaka-type duality, and a way to extract characters of representations from $K$-theory. Finally, we obtain a continuity property near the identity element of $G$ of families of maps $K_0(C^*_rG)\to \mathbb{C}$, parametrised by semisimple elements of $G$, defined by stable orbital integrals. This implies a continuity property for $L$-packets of discrete series characters, which in turn can be used to deduce a (well-known) expression for formal degrees of discrete series representations from Harish-Chandra's character formula.

Some combinatorial identities appearing in the calculation of the cohomology of Siegel modular varieties

In the computation of the intersection cohomology of Shimura varieties, or of the L 2 cohomology of equal rank locally symmetric spaces, combinatorial identities involving averaged discrete series characters of real reductive groups play a large technical role. These identities can become very complicated and are not always well-understood (see for example the appendix of [8]). We propose a geometric approach to these identities in the case of Siegel modular varieties using the combinatorial properties of the Coxeter complex of the symmetric group. Apart from some introductory remarks about the origin of the identities, our paper is entirely combinatorial and does not require any knowledge of Shimura varieties or of representation theory.

A uniform classification of discrete series representations of affine Hecke algebras

We give a new and independent parameterization of the set of discrete series characters of an affine Hecke algebra $\mathcal{H}_{\mathbf{v}}$, in terms of a canonically defined basis $\mathcal{B}_{gm}$ of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras $\mathcal{H}$, and to all $\mathbf{v}\in\mathcal{Q}$, where $\mathcal{Q}$ denotes the vector group of positive real (possibly unequal) Hecke parameters for $\mathcal{H}$. By analytic Dirac induction we define for each $b\in \mathcal{B}_{gm}$ a continuous (in the sense of [OS2]) family $\mathcal{Q}^{reg}_b:=\mathcal{Q}_b\backslash\mathcal{Q}_b^{sing}\ni\mathbf{v}\to\operatorname{Ind}_{D}(b;\mathbf{v})$, such that $\epsilon(b;\mathbf{v})\operatorname{Ind}_{D}(b;\mathbf{v})$ (for some $\epsilon(b;\mathbf{v})\in\{\pm 1\}$) is an irreducible discrete series character of $\mathcal{H}_{\mathbf{v}}$. Here $\mathcal{Q}^{sing}_b\subset\mathcal{Q}$ is a finite union of hyperplanes in $\mathcal{Q}$. In the non-simply laced cases we show that the families of virtual discrete series characters $\operatorname{Ind}_{D}(b;\mathbf{v})$ are piecewise rational in the parameters $\mathbf{v}$. Remarkably, the formal degree of $\operatorname{Ind}_{D}(b;\mathbf{v})$ in such piecewise rational family turns out to be rational. This implies that for each $b\in \mathcal{B}_{gm}$ there exists a universal rational constant $d_b$ determining the formal degree in the family of discrete series characters $\epsilon(b;\mathbf{v})\operatorname{Ind}_{D}(b;\mathbf{v})$. We will compute the canonical constants $d_b$, and the signs $\epsilon(b;\mathbf{v})$. For certain geometric parameters we will provide the comparison with the Kazhdan-Lusztig-Langlands classification.

Spectral transfer morphisms for unipotent affine Hecke algebras

We classify the spectral transfer morphisms (cf. Opdam in Adv Math 286:912–957, 2016) between affine Hecke algebras associated to the unipotent types of the various inner forms of an unramified absolutely simple algebraic group G defined over a non-archimedean local field k. This turns out to characterize Lusztig’s classification (Lusztig in Int Math Res Not 11:517–589, 1995; in Represent Theory 6:243–289, 2002) of unipotent characters of G in terms of the Plancherel measure, up to diagram automorphisms. As an application of these results, the spectral correspondences associated with such morphisms (Opdam 2016), and some results of Ciubotaru, Kato and Kato [CKK] (also see Ciubotaru and Opdam in A uniform classification of the discrete series representations of affine Hecke algebras. arXiv:1510.07274) we prove a conjecture of Hiraga, Ichino and Ikeda [HII] on formal degrees and adjoint gamma factors in the special case of unipotent discrete series characters of inner forms of unramified simple groups of adjoint type defined over k.

Discrete series characters for affine Hecke algebras and their formal degrees

We introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types ${E_{n}^{(1)}}$ , n=6, 7, 8) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases).

Stable Discrete Series Characters at Singular Elements

AbstractWrite for the stable discrete series character associated with an irreducible finite-dimensional representation E of a connected real reductive group G. Let M be the centralizer of the split component of a maximal torus T, and denote by Arthur’s extension of . In this paper we give a simple explicit expression for when γ is elliptic in G. We do not assume γ is regular.

Homological algebra for affine Hecke algebras

In this paper we study homological properties of modules over an affine Hecke algebra H . In particular we prove a comparison result for higher extensions of tempered modules when passing to the Schwartz algebra S , a certain topological completion of the affine Hecke algebra. The proof is self-contained and based on a direct construction of a bounded contraction of certain standard resolutions of H -modules. This construction applies for all positive parameters of the affine Hecke algebra. This is an important feature, since it is an ingredient to analyse how the irreducible discrete series representations of H arise in generic families over the parameter space of H . For irreducible non-simply laced affine Hecke algebras this will enable us to give a complete classification of the discrete series characters, for all positive parameters (we will report on this application in a separate article).

Stability in level 0, for the odd orthogonal $p$-adic groups

The general problem we discuss in this paper is how to prove stability properties for a linear combination of characters of irreductible discrete series of p-adic groups. Here we give ideas on how to reduce the case where the Langlands parameter is trivial on the wild ramification group to the case where this Langlands parameter factorizes through the Frobenius; we handle only the case of an odd orthogonal group. The principal result is that the localization commutes with the Lusztig's induction.

Discrete Series Characters as Lifts from Two-structure Groups

Let G G be a connected reductive Lie group with a relatively compact Cartan subgroup. Then it has relative discrete series representations. The main result of this paper is a formula expressing relative discrete series characters on G G as “lifts” of relative discrete series characters on smaller groups called two-structure groups for G G . The two-structure groups are connected reductive Lie groups which are locally isomorphic to the direct product of an abelian group and simple groups which are real forms of S L ( 2 , C ) SL(2, \mathbf {C}) or S O ( 5 , C ) SO(5, \mathbf {C} ) . They are not necessarily subgroups of G G , but they “share” the relatively compact Cartan subgroup and certain other Cartan subgroups with G G . The character identity is similar to formulas coming from endoscopic lifting, but the two-structure groups are not necessarily endoscopic groups, and the characters lifted are not stable. Finally, the formulas are valid for non-linear as well as linear groups.

Stable discrete series characters as lifts from complex two-structure groups

Stable discrete series characters as lifts from complex two-structure groups

Modular invariance of string theory on AdS3

We discuss the modular invariance of the SL(2, R) WZW model. In particular, we discuss in detail the modular invariants using the sl ̂ (2,R) characters based on the discrete unitary series of the SL(2, R) representations. The explicit forms of the corresponding characters are known when no singular vectors appear. We show, for example, that from such characters modular invariants can be obtained only when the level k<2 and infinitely large spins are included. In fact, we give a modular invariant with three variables Z( z, τ, u) in this case. We also argue that the discrete series characters are not sufficient to construct a modular invariant compatible with the unitarity bound, which was proposed to resolve the ghost problem of the SL(2, R) strings.

The characters of the generalized Steinberg representations of finite general linear groups on the regular elliptic set

Let k be a finite field, k Ik the degree n extension of k, and G = GLn(k) the general linear group with entries in k. This paper studies the generalized Steinberg (GS) representationis of G and proves the equivalence of several different characterizations for this class of representations. As our main result we show that the union of the class of cuspidal and GS representations of G is in natural one-one correspondence with the set of Galois orbits of characters of kx, the regular orbits of course corresponding to the cuspidal representations. Besides using Green's character formulas to define GS representations, we characterize GS representations by associating to them idempotents in certain commuting algebras corresponding to parabolic inductions and by showing that GS representations are the sole components of these induced representations which are generic (have Whittaker vectors). Let k = Fq be a finite field of cardinality q, let kIkm Ik be,_ respectively, an algebraic closure of k and the degree rm extension of k contained in k. Let q x _ Xq denote the Frobenius automorphism of klk and of any subextension km1k. Let G = GL (k) be the group of non-singular n x n matrices with entries in k. Also write Gm = GLrn(k) for any m > 1. In a signally important paper published in 1955 [GR] J. A. Green showed how to calculate formulas for the irreducible characters of G. In Green's work appeared for the first time general character formulas for the cuspidal representations and for a family of generalized Steinberg (GS) representations. For their study of the level zero discrete series characters of unit groups of simple algebras over a p-adic field the authors need diverse characterizations of cuspidal and GS characters of finite general linear groups and to be able to pass between these different characterizations. In this paper we give these characterizar tions and prove their equivalence. It is fruitful to view the set of GS representatio:ns as a class of representations of G which contains the class of cuspidal representations as a subelass. Of course the cusp form property, the propert;y of not being a component of Ind u 1 for any unipotent radical U :& (I) of a parabolic subgroup of G, clearly distinguishes the class of cuspidal representations from all other representations of G. However, other important properties which are usually associated to the class of cuspidal representations generalize to the union of the two classes of representations of C. Received by the editors May 26, 1997 and, in revised form, April 18, 1998 and June 26, 1998. 1991 Mathematics Subject Classification. Primary 22E50, 11T24.

Discrete Series Characters for GL(n, q)

The ‘discrete series’ characters of the finite general linear group GL(n, q) are expressed as uniquely defined integral linear combinations of characters induced from linear characters on certain subgroups Hd, n of GL(n, q). The coefficients in these linear combinations are determined (for all n, q) by a family of polynomials rλ(T) ∈ Z[T] indexed by the set of all partitions λ.

Correction to “Discrete series characters and the Lefschetz formula for Hecke operators”

Correction to “Discrete series characters and the Lefschetz formula for Hecke operators”

Discrete series characters and two-structures

Let G be a connected semisimple real Lie group with compact Cartan subgroup. Harish-Chandra gave formulas for discrete series characters which are completely explicit except for certain interger constants appearing in the numerators. The main result of this paper is a new formula for these constants using two-structures. The new formula avoids endoscopy and stable discrete series entirely, expressing (unaveraged) discrete series constants directly in terms of (unaveraged) discrete series constants corresponding to two-structures of noncompact type.

Discrete series characters and the Lefschetz formula for Hecke operators

This paper consists of three independent but related parts. In the rst part (xx1{ 6) we give a combinatorial formula for the constants appearing in the \numerators of characters of stable discrete series representations of real groups (see x3) as well as an analogous formula for individual discrete series representations (see x6). Moreover we give an explicit formula (Theorem 5.1) for certain stable virtual characters on real groups; by Theorem 5.2 these include the stable discrete series characters, and thus we recover the results of x3 in a more natural way. In the second part (x7) we use the character formula given in Theorem 5.1 to rewrite the Lefschetz formula of [GM] (for the local contribution at a single xed point component to the trace of a Hecke operator on weighted cohomology) in the same spirit as that of Arthur's Lefschetz formula [A]: in terms of stable virtual characters on real groups (see Theorem 7.14.B). We then sum the contributions of the various xed point components and show that, in the case of middle weighted cohomology, the resulting global Lefschetz xed point formula agrees with Arthur's Lefschetz formula. This gives a topological proof of Arthur's formula.