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:
| [1] | Diamond, Fred, A correspondence between representations of local Galois groups and Lie-type groups, (L-Functions and Galois Representations. L-Functions and Galois Representations, London Math. Soc. Lecture Note Ser., vol. 320 (2007), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 187-206, MR 2392355 · Zbl 1230.11069 |
| [2] | Gelfand, S. I., Representations of the full linear group over a finite field, Mat. Sb. (N.S.), 83, 125, 15-41 (1970), MR 0272916 |
| [3] | Gross, Benedict H., Some applications of Gelfand pairs to number theory, Bull. Am. Math. Soc. (N.S.), 24, 2, 277-301 (1991), MR 1074028 · Zbl 0733.11018 |
| [4] | Kawanaka, Noriaki, On the irreducible characters of the finite unitary groups, J. Math. Soc. Jpn., 29, 3, 425-450 (1977), MR 450383 · Zbl 0353.20031 |
| [5] | Macdonald, I. G., Zeta functions attached to finite general linear groups, Math. Ann., 249, 1, 1-15 (1980), MR 575444 · Zbl 0417.14016 |
| [6] | Mao, Zhengyu, Spherical Bessel functions of \(\operatorname{GL}_2( \mathbf{F}_{q^2})\), J. Number Theory, 87, 1, 154-171 (2001), MR 1816041 · Zbl 1034.11066 |
| [7] | Piatetski-Shapiro, Ilya, Complex Representations of \(\operatorname{GL}(2, K)\) for Finite Fields K, Contemporary Mathematics, vol. 16 (1983), American Mathematical Society: American Mathematical Society Providence, R.I., MR 696772 · Zbl 0513.20026 |
| [8] | Shintani, Takuro, Two remarks on irreducible characters of finite general linear groups, J. Math. Soc. Jpn., 28, 2, 396-414 (1976), MR 414730 · Zbl 0323.20041 |
| [9] | Tate, J., Number theoretic background, (Automorphic Forms, Representations and L-Functions, Part 2, Proc. Sympos. Pure Math.. Automorphic Forms, Representations and L-Functions, Part 2, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977. Automorphic Forms, Representations and L-Functions, Part 2, Proc. Sympos. Pure Math.. Automorphic Forms, Representations and L-Functions, Part 2, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977, Proc. Sympos. Pure Math., vol. XXXIII (1979), Amer. Math. Soc.: Amer. Math. Soc. Providence, R.I.), 3-26, MR 546607 · Zbl 0422.12007 |
| [10] | Vatsal, Vinayak, Test vectors for some ramified representations (2018) · Zbl 1367.11051 |
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