A note on rationality of orbital integrals on a \(p\)-adic group. (English) Zbl 0856.22019
Let \(F\) be a non-Archimedean field with valuation function val: \(F\to\mathbb{Z}\cup \{\infty\}\), the ring of integers \(O_F\) and the maximal ideal \(P_F\). Let \(q\) be the cardinality of the residue field \(O_F/P_F\). Choose a uniformizer \(\widetilde \omega\) in \(O_F\) with \(|\widetilde \omega|= q^{- 1}\). Put \(\text{ac}(x)= x \widetilde \omega^{- \text{val}(x)}\). Given \(f\in F[x]\), \(f\not\in F\), a locally constant function \(\Phi: F^n \to \mathbb{C}\) with compact support, and a character \(\chi\) of \(O^X_F\) (with \(\chi(0)= 0\)), the local Igusa zeta function is defined by
\[
Z_\Phi(s, \chi, f)= \int_{F^n} \Phi(x) \chi(\text{ac}(f(x)))|f(x)|^s\cdot |dx|.
\]
By a regular prehomogeneous vector space (p.v.s.) over \(F\) we mean a triple \((G, \rho, V)\) wherein: (i) \(G\) is a reductive algebraic group defined over \(F\); (ii) \(V\) is a finite-dimensional vector space over \(F\); (iii) \(\rho\) is a rational representation of \(G\) on \(V\); (iv) there is a proper algebraic set \(S\subset V\) such that \(V- S\) constitutes a single \(G\)-orbit; (v) the isotropy subgroup \(G_x\) of any \(x\in V- S\) is reductive.
Given a rational character \(\chi\) of \(G\) defined over \(F\), then a nonzero rational function \(f\) on \(V\), such that \(f(\rho(g)\cdot x)= \chi(g) f(x)\) for all \(g\) in \(G\) and all \(x\) in \(V\), is said to be a relative invariant with respect to \(\chi\) for \((G, \rho, V)\). Let \(\Omega\) denote an open \(G(F)\)-orbit in \(V(F)\), \(\varphi\) a non-trivial relative invariant, \(\Phi\) a Schwartz-Bruhat function on \(V(F)\). The local orbital zeta function \(Z_\Phi(\Omega, \varphi, s)\) is defined by \[ Z_\Phi(\Omega, \varphi, s)= \int_\Omega \Phi(x) |\varphi(x)|^s |dx|. \] \(Z_\Phi\) is a rational function of \(q^{- s}\) (Igusa’s theorem).
In this paper, the author points out interesting connections between unipotent orbital integrals and Igusa local zeta functions (Theorem 3): For any \(x\in G(F)\) and any fixed \(E\equiv \mathbb{C}\) the following holds: \(f\in C^\infty_c(G(F))\) and \(f(y)\in E\) for all \(y\in F\Rightarrow \int_{O(x)} f\in E\). (Theorem 4): For each unipotent orbit \(O\) in \(G(F)\) the associated Shalika germ is a rational valued function. Hence it follows, in certain cases, that the coefficients appearing in the Harish-Chandra local character expansion are rational numbers.
Given a rational character \(\chi\) of \(G\) defined over \(F\), then a nonzero rational function \(f\) on \(V\), such that \(f(\rho(g)\cdot x)= \chi(g) f(x)\) for all \(g\) in \(G\) and all \(x\) in \(V\), is said to be a relative invariant with respect to \(\chi\) for \((G, \rho, V)\). Let \(\Omega\) denote an open \(G(F)\)-orbit in \(V(F)\), \(\varphi\) a non-trivial relative invariant, \(\Phi\) a Schwartz-Bruhat function on \(V(F)\). The local orbital zeta function \(Z_\Phi(\Omega, \varphi, s)\) is defined by \[ Z_\Phi(\Omega, \varphi, s)= \int_\Omega \Phi(x) |\varphi(x)|^s |dx|. \] \(Z_\Phi\) is a rational function of \(q^{- s}\) (Igusa’s theorem).
In this paper, the author points out interesting connections between unipotent orbital integrals and Igusa local zeta functions (Theorem 3): For any \(x\in G(F)\) and any fixed \(E\equiv \mathbb{C}\) the following holds: \(f\in C^\infty_c(G(F))\) and \(f(y)\in E\) for all \(y\in F\Rightarrow \int_{O(x)} f\in E\). (Theorem 4): For each unipotent orbit \(O\) in \(G(F)\) the associated Shalika germ is a rational valued function. Hence it follows, in certain cases, that the coefficients appearing in the Harish-Chandra local character expansion are rational numbers.
Reviewer: T.S.Bhanu Murthy (Madras)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 22E35 | Analysis on \(p\)-adic Lie groups |
Keywords:
Igusa’s theorem; non-Archimedean field; Igusa zeta function; reductive algebraic group; rational representation; character; Schwartz-Bruhat function; local orbital zeta function; orbital integrals; Igusa local zeta functions; Shalika germ; Harish-Chandra local character expansionReferences:
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