Orthogonality of invariant vectors. (English) Zbl 1512.20020
Summary: Let \(G\) be a finite group with given subgroups \(H\) and \(K\). Let \(\pi\) be an irreducible complex representation of \(G\) such that its space of \(H\)-invariant vectors as well as the space of \(K\)-invariant vectors are both one dimensional. Let \(v_H\) (resp. \( v_K)\) denote an \(H\)-invariant (resp. \(K\)-invariant) vector of unit norm in a given \(G\)-invariant inner product \(\langle, \rangle_\pi\) on \(\pi \). Our interest is in computing the square of the absolute value of \(\langle v_H, v_K \rangle_\pi \). This is the correlation constant \(c(\pi; H, K)\) defined by B. H. Gross [Bull. Am. Math. Soc., New Ser. 24, No. 2, 277–301 (1991; Zbl 0733.11018)]. In this paper, we study this question for \(G = \operatorname{GL}_2( \mathbb{F}_q)\), where \(\mathbb{F}_q\) is the finite field of \(q = p^m\) elements of odd characteristic \(p, H\) is its split torus and \(K\) is a non-split torus. The first main theorem of this paper gives an explicit formula for \(|\langle v_H, v_K \rangle_\pi|^2\) modulo \(p\). The key idea here is to analyse the mod \(p\) reduction of \(\pi \). The second main theorem relates the behaviour of \(\langle v_H, v_K \rangle_\pi\) under Shintani base change and gives a sufficient condition for \(\langle v_H, v_K \rangle_\pi\) to vanish for an irreducible representation \(\pi = \operatorname{BC}(\tau)\) of \(\operatorname{PGL}_2(\mathbb{E})\), in terms of the epsilon factor of the base changing representation \(\tau\) of \(\operatorname{PGL}_2(\mathbb{F})\), where \(\mathbb{E} / \mathbb{F}\) is a finite extension of finite fields.
MSC:
20C15 | Ordinary representations and characters |
20C20 | Modular representations and characters |
20C33 | Representations of finite groups of Lie type |
20G05 | Representation theory for linear algebraic groups |
20G40 | Linear algebraic groups over finite fields |
Citations:
Zbl 0733.11018References:
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