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There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio trace(pi(g)) / dim(pi), for an irreducible representation pi of G and an element g of G. It turns out [Gurevich-Howe15, Gurevich-Howe17] that for classical groups G over finite fields there are several (compatible) invariants of representations that provide strong information on the character ratios. We call these invariants collectively rank. Rank suggests a new way to organize the representations of classical groups over finite and local fields - a way in which the building blocks are the smallest representations. This is in contrast to Harish-Chandras philosophy of cusp forms that is the main organizational principle since the 60s, and in it the building blocks are the cuspidal representations which are, in some sense, the LARGEST. The philosophy of cusp forms is well adapted to establishing the Plancherel formula for reductive groups over local fields, and led to Lusztigs classification of the irreducible representations of such groups over finite fields. However, analysis of character ratios might benefit from a different approach. In this note we discuss further the notion of tensor rank for GL_n over a finite field F_q and demonstrate how to get information on representations of a given tensor rank using tools coming from the recently studied eta correspondence, as well as the well known philosophy of cusp forms, mentioned just above. A significant discovery so far is that although the dimensions of the irreducible representations of a given tensor rank vary by quite a lot (they can differ by large powers of q), for certain group elements of interest the character ratios of these irreps are nearly equal to each other.
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In [Frobenius1896] it was shown that many important properties of a finite group could be examined using formulas involving the character ratios of group elements, i.e., the trace of the element acting in a given irreducible representation, divided by the dimension of the representation. In [Gurevich-Howe15] and [Gurevich-Howe17], the current authors introduced the notion of rank of an irreducible representation of a finite classical group. One of the motivations for studying rank was to clarify the nature of character ratios for certain elements in these groups. In fact in the above cited papers, two notions of rank were given. The first is the Fourier theoretic based notion of U-rank of a representation, which comes up when one looks at its restrictions to certain abelian unipotent subgroups. The second is the more algebraic based notion of tensor rank which comes up naturally when one attempts to equip the representation ring of the group with a grading that reflects the central role played by the few smallest possible representations of the group. In [Gurevich-Howe17] we conjectured that the two notions of rank mentioned just above agree on a suitable collection called low rank representations. In this note we review the development of the theory of rank for the case of the general linear group GL_n over a finite field F_q, and give a proof of the agreement conjecture that holds true for sufficiently large q. Our proof is Fourier theoretic in nature, and uses a certain curious positivity property of the Fourier transform of the set of matrices of low enough fixed rank in the vector space of matrices of size m x n over F_q. In order to make the story we are trying to tell clear, we choose in this note to follow a particular example that shows how one might apply the theory of rank to certain counting problems.
Let $mathsf k$ be a local field. Let $I_ u$ and $I_{ u}$ be smooth principal series representations of $mathrm{GL}_n(mathsf k)$ and $mathrm{GL}_{n-1}(mathsf k)$ respectively. The Rankin-Selberg integrals yield a continuous bilinear map $I_ utimes I_{ u}rightarrow mathbb C$ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map $I_ utimes I_{ u}rightarrow mathbb C$ with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for $mathrm{GL}_n(mathsf k)times mathrm{GL}_n(mathsf k)$.
The Pieri rule gives an explicit formula for the decomposition of the tensor product of irreducible representation of the complex general linear group GL(n,C) with a symmetric power of the standard representation on C^n. It is an important and long understood special case of the Littlewood-Richardson rule for decomposing general tensor products of representations of GL(n,C). In our recent work [Gurevich-Howe17, Gurevich-Howe19] on the organization of representations of the general linear group over a finite field F_q using small representations, we used a generalization of the Pieri rule to the context of this latter group. In this note, we demonstrate how to derive the Pieri rule for GL(n,Fq). This is done in two steps; the first, reduces the task to the case of the symmetric group S_n, using the natural relation between the representations of S_n and the spherical principal series representations of GL(n,F_q); while in the second step, inspired by a remark of Nolan Wallach, the rule is obtained for S_n invoking the S_ell-GL_(n,C)) Schur duality. Along the way, we advertise an approach to the representation theory of the symmetric group which emphasizes the central role played by the dominance order on Young diagrams. The ideas leading to this approach seem to appear first, without proofs, in [Howe-Moy86].
Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra $mathfrak{gl}_{m|n}(mathbb{C})$. In this paper we study the {em Whittaker coinvariants functor}, which is an exact functor from category $mathcal O$ for $mathfrak{gl}_{m|n}(mathbb{C})$ to a certain category of finite-dimensional modules over $W_{m|n}$. We show that this functor has properties similar to Soergels functor $mathbb V$ in the setting of category $mathcal O$ for a semisimple Lie algebra. We also use it to compute the center of $W_{m|n}$ explicitly, and deduce some consequences for the classification of blocks of $mathcal O$ up to Morita/derived equivalence.
We study multiplicities of unipotent characters in tensor products of unipotent characters of GL(n,q). We prove that these multiplicities are polynomials in q with non-negative integer coefficients. We study the degree of these polynomials and give a necessary and sufficient condition in terms of the representation theory of symmetric groups for these polynomials to be non-zero.
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