[Paper Review] Weak Gelfand Pair Property And Application To GL(n+1),GL(n) Over Finite Fields

Yoav Ben Shalom|arXiv (Cornell University)|Oct 28, 2012

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper establishes that for GL(n+1, F_q) and GL(n, F_q) over finite fields, every irreducible representation has at most two GL(n, F_q)-invariant vectors, proving a weak Gelfand pair property via a generalized Gelfand trick using the transpose anti-involution. The key result is dim π^{GL_n(F_q)} ≤ 2 for all irreducible π over algebraically closed fields of characteristic ≠ 2, with a similar bound for orthogonal groups when q is not a power of 2.

ABSTRACT

Let F_q be the finite field with q elements. Consider the standard embedding GL(n,F_q) -> GL(n+1,F_q). In this paper we prove that for every irreducible representation pi of GL(n+1,F_q) over algebraically closed fields of characteristic different from 2 we have dimπ^GL(n,F_q)<=2. To do that we define a property of weak Gelfand pair and prove a generalization of Gelfand trick for weak Gelfand pairs, using the anti-involution transpose to get the result for GL(n+1,F_q),GL(n,F_q). In a similar manner we show that for q not a power of 2 O(n+1,F_q),O(n,F_q) is a Gelfand pair over algebraically closed fields of characteristic different from 2.

Motivation & Objective

• To quantify the failure of the GL(n+1, F_q), GL(n, F_q) pair to be a Gelfand pair over finite fields.
• To define and develop the theory of weak Gelfand pairs as a generalization of classical Gelfand pairs.
• To extend the Gelfand trick to settings where the anti-involution does not preserve all double-cosets, allowing bounds on multiplicity of invariant vectors.
• To prove that (O_{n+1}(F_q), O_n(F_q)) is a Gelfand pair when q is not a power of 2, over fields of characteristic ≠ 2.
• To provide a representation-theoretic framework applicable over arbitrary algebraically closed fields of characteristic ≠ 2, not just C.

Proposed method

• Introduce the concept of a weak Gelfand pair, defined by a bound on the dimension of H-invariant vectors in irreducible G-representations.
• Generalize the classical Gelfand trick using an anti-involution σ that preserves H and the central character of π, with a bound on the number of non-preserved double-cosets.
• Use the transpose map as the anti-involution σ on GL(n+1, F_q), which preserves GL(n, F_q) and the central character of irreducible representations.
• Prove that only two double-cosets in GL_n(F_q)\GL_{n+1}(F_q)/Z(GL_{n+1}(F_q))GL_n(F_q) are not preserved under transpose, leading to k = 1 in the bound.
• Apply Lemma 2.2 to deduce dim π^{GL_n(F_q)} ≤ k + 1 = 2 for all irreducible π over F with char(F) ≠ 2.
• Extend the method to orthogonal groups using the same anti-involution and double-coset counting, showing that (O_{n+1}(F_q), O_n(F_q)) is a Gelfand pair when q is not a power of 2.

Experimental results

Research questions

• RQ1 What is the maximum possible dimension of the space of GL(n, F_q)-invariant vectors in an irreducible representation of GL(n+1, F_q) over fields of characteristic ≠ 2?
• RQ2 Can the classical Gelfand trick be generalized to cases where the anti-involution does not preserve all double-cosets?
• RQ3 How does the number of non-preserved double-cosets under transpose affect the multiplicity of invariant vectors?
• RQ4 Under what conditions is the pair (O_{n+1}(F_q), O_n(F_q)) a Gelfand pair over finite fields?
• RQ5 Can the weak Gelfand pair property be used to derive bounds on multiplicity in representations of classical groups over finite fields?

Key findings

• For every irreducible representation π of GL_{n+1}(F_q) over an algebraically closed field F with char(F) ≠ 2, the dimension of the space of GL_n(F_q)-invariant vectors satisfies dim π^{GL_n(F_q)} ≤ 2.
• The number of GL_n(F_q)-double-cosets in GL_{n+1}(F_q) (modulo the center) that are not preserved under transpose is exactly two, leading to k = 1 in the bound.
• The generalized Gelfand trick using transpose as anti-involution yields the bound dim π^H ≤ k + 1 = 2, where k is the number of non-preserved double-cosets divided by 2.
• For q not a power of 2, the pair (O_{n+1}(F_q), O_n(F_q)) is a Gelfand pair over F with char(F) ≠ 2, as the transpose anti-involution preserves all double-cosets.
• The result holds over arbitrary algebraically closed fields of characteristic ≠ 2, not just C, extending the scope of classical results.
• The proof relies on the existence of symmetric matrices in GL_n(F_q) that map any non-zero vector to any other non-zero vector, ensuring transitivity on double-coset representatives.