Representations Of General Linear Groups Research Articles

An inductive approach to representations of general linear groups over compact discrete valuation rings

In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called PSH-algebras. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bialgebra that contains a large PSH-algebra that extends Zelevinsky's algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.

A representation-theoretic interpretation of positroid classes

A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology class Poincaré dual to a positroid variety. We define a family of representations of general linear groups whose characters are symmetric polynomials representing positroid classes. These representations are certain diagram Schur modules in the sense of James and Peel. This gives a new algebraic interpretation of the Schubert structure constants for the product of a Schubert polynomial and Schur polynomial, and of the 3-point Gromov-Witten invariants for Grassmannians, proving a conjecture of Postnikov. As a byproduct, we obtain an effective algorithm for decomposing positroid classes into Schubert classes.

A triangular system for local character expansions of Iwahori-spherical representations of general linear groups

For Iwahori-spherical representations of non-Archimedean general linear groups, Chan–Savin recently expressed the Whittaker functor as a restriction to an isotypic component of a finite Iwahori–Hecke algebra module. We generalize this method to describe principal degenerate Whittaker functors. Concurrently, we view Murnaghan’s formula for the Harish-Chandra–Howe character as a Grothendieck group expansion of the same module.

Representations of General Linear Groups in the Verlinde Category

In this article, we construct affine group schemes GL(X) where X is any object in the Verlinde category in characteristic p and classify their irreducible representations. We begin by showing that for a simple object X of categorical dimension i, this representation category is semisimple and is equivalent to the connected component of the Verlinde category for SLi. Subsequently, we use this along with a Verma module construction to classify irreducible representations of GL(nL) for any simple object L and any natural number n. Finally, parabolic induction allows us to classify irreducible representations of GL(X) where X is any object in Verp.

Inner functions of matrix argument and conjugacy classes in unitary groups

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.

Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)

Abstract We prove a local Gan–Gross–Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean fields. We also generalize to Bessel and Fourier–Jacobi models and study a possible generalization to Ext-branching laws.

On restriction of unitarizable representations of general linear groups and the non-generic local Gan–Gross–Prasad conjecture

We prove one direction of a recently posed conjecture by Gan-Gross-Prasad, which predicts the branching laws that govern restriction from p-adic $GL_n$ to $GL_{n-1}$ of irreducible smooth representations within the Arthur-type class. We extend this prediction to the full class of unitarizable representations, by exhibiting a combinatorial relation that must be satisfied for any pair of irreducible representations, in which one appears as a quotient of the restriction of the other. We settle the full conjecture for the cases in which either one of the representations in the pair is generic. The method of proof involves a transfer of the problem, using the Bernstein decomposition and the quantum affine Schur-Weyl duality, into the realm of quantum affine algebras. This restatement of the problem allows for an application of the combined power of a result of Hernandez on cyclic modules together with the Lapid-Minguez criterion from the p-adic setting.

Robinson–Schensted–Knuth correspondence in the representation theory of the general linear group over a non-archimedean local field

We construct new “standard modules” for the representations of general linear groups over a local non-archimedean field. The construction uses a modified Robinson–Schensted–Knuth correspondence for Zelevinsky’s multisegments. Typically, the new class categorifies the basis of Doubilet, Rota, and Stein (DRS) for matrix polynomial rings, indexed by bitableaux. Hence, our main result provides a link between the dual canonical basis (coming from quantum groups) and the DRS basis.

Multiplicities under basechange: Finite field case

A general proposition is proved relating multiplicities (for the restriction of a representation of a group to a subgroup) under basechange for groups over finite fields, and used to calculate certain multiplicities for cuspidal representations of general linear groups which become principal series representations under basechange for which the multiplicities can be calculated by geometric methods. For E/F quadratic extension of finite fields, we use this method to calculate which 1 dimensional representations of GLn(E) appear in a cuspidal representation of GL2n(F). We also calculate which 1 dimensional representations of GLn(F)×GLn(F) appear in a cuspidal representation of GL2n(F).

An automorphic generalization of the Hermite–Minkowski theorem

We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,1,...,23\}$. More generally, we define a simple sequence $(r(w))_{w \geq 0}$ such that for any integer $w$, any number field $E$ whose root-discriminant is less than $r(w)$, and any ideal $N$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of general linear groups over $E$ whose conductor is $N$ and whose weights are in the interval $\{0,1,...,w\}$. Assuming a version of GRH, we also show that we may replace $r(w)$ with $8 \pi e^{\gamma-H_w}$ in this statement, where $\gamma$ is Euler's constant and $H_w$ the $w$-th harmonic number. The proofs are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin-Selberg $L$-functions. Both the effectiveness and the optimality of the methods are discussed.

Arthur parameters and cuspidal automorphic modules of classical groups

The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks: How to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well-expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.

Editorial

Editorial

James Alexander Green. 26 February 1926—7 April 2014

James Alexander Green, known as Sandy, was a mathematician of great influence and distinction. He was an algebraist, famous for his work on modular representations of finite groups, and the development of the theory of polynomial representations of general linear groups. He was elected Fellow of the Royal Society of Edinburgh (1968) and Fellow of the Royal Society of London (1987). He was awarded prizes of the London Mathematical Society, a Senior Berwick Prize (in 1984) and the De Morgan Medal (in 2001). In his doctoral thesis, on semigroups, Sandy introduced fundamental relations, now known as ‘Green's relations’. He determined the characters of arbitrary finite general linear groups published 1955. Sandy then turned to representations of finite groups over fields of prime characteristic; his work laid the foundations for the module theoretic approach to the subject. His next highlight is his monograph on polynomial representations of GL n , published in 1980, which has become the basis for algebraic highest weight theory. Furthermore, in 1995 he proved a fundamental result on Hall algebras, establishing a connection between quantum groups and representations of finite-dimensional quiver algebras.

The unipotent modules of $${{\mathrm {GL}}}_{n}({{\mathbb {F}}}_{q})$$ GL n ( F q ) via tableaux

We construct the irreducible unipotent modules of the finite general linear groups from actions on tableaux. Our approach is analogous to that of James (Bull Lond Math Soc 8:229–232, 1976) for the symmetric groups, answering an open question as to whether such a construction exists. We show that our modules are isomorphic to those previously constructed by James (Representations of general linear groups, London Mathematical Society Lecture Note Series, vol. 94. Cambridge University Press, Cambridge, 1984. doi:10.1017/CBO9780511661921) , although the two presentations are quite different. Key to our construction are the generalized Gelfand–Graev representations of Kawanaka (Generalized Gel’fand-Graev representations and Ennola duality. In: Algebraic groups and related topics (Kyoto/Nagoya, 1983), advanced studies in pure math., vol. 6, pp. 175–206. North-Holland, Amsterdam 1985).

Local symmetric square L-factors of representations of general linear groups

Local symmetric square L-factors of representations of general linear groups

COMPOSITION FACTORS OF A CLASS OF INDUCED REPRESENTATIONS OF CLASSICAL -ADIC GROUPS

We study induced representations of the form $\unicode[STIX]{x1D6FF}_{1}\times \unicode[STIX]{x1D6FF}_{2}\rtimes \unicode[STIX]{x1D70E}$, where $\unicode[STIX]{x1D6FF}_{1},\unicode[STIX]{x1D6FF}_{2}$ are irreducible essentially square-integrable representations of general linear group and $\unicode[STIX]{x1D70E}$ is a strongly positive discrete series of classical $p$-adic group, which naturally appear in the nonunitary dual. For $\unicode[STIX]{x1D6FF}_{1}=\unicode[STIX]{x1D6FF}([\unicode[STIX]{x1D708}^{a}\unicode[STIX]{x1D70C}_{1},\unicode[STIX]{x1D708}^{b}\unicode[STIX]{x1D70C}_{1}])$ and $\unicode[STIX]{x1D6FF}_{2}=\unicode[STIX]{x1D6FF}([\unicode[STIX]{x1D708}^{c}\unicode[STIX]{x1D70C}_{2},\unicode[STIX]{x1D708}^{d}\unicode[STIX]{x1D70C}_{2}])$ with $a\geqslant 1$ and $c\geqslant 1$, we determine composition factors of such induced representation.

Polynomial representations of $$\mathrm{GL }(n)$$ GL ( n ) and Schur–Weyl duality

Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur–Weyl duality is the key for an equivalence between both categories.

Highest weight $$\mathfrak{sl }_2$$ -categorifications I: crystals

We define highest weight categorical actions of \(\mathfrak{sl }_2\) on highest weight categories and show that basically all known examples of categorical \(\mathfrak{sl }_2\)-actions on highest weight categories (including rational and polynomial representations of general linear groups, parabolic categories \(\mathcal O \) of type \(A\), categories \(\mathcal O \) for cyclotomic Rational Cherednik algebras) are highest weight in our sense. Our main result is an explicit combinatorial description of (the labels of) the crystal on the set of simple objects. A new application of this is to determining the supports of simple modules over the cyclotomic Rational Cherednik algebras starting from their labels.

Geometric interpretation of Murphy bases and an application

In this article we study the representations of general linear groups which arise from their action on flag spaces. These representations can be decomposed into irreducibles by proving that the associated Hecke algebra is cellular. We give a geometric interpretation of a cellular basis of such Hecke algebras which was introduced by Murphy in the case of finite fields. We apply these results to decompose representations which arise from the space of submodules of a free module over principal ideal local rings of length two with a finite residue field.

On representations of general linear groups over principal ideal local rings of length two

We study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. For general linear groups of order three and four over these rings, we construct all the irreducible representations. We show that the problem of constructing all the irreducible representations of all general linear groups over these rings is not easier than the problem of constructing all the irreducible representations of all general linear groups over principal ideal local rings of arbitrary length in the function field case.