Trace paley-wiener theorem for reductivep-adic groups
Trace paley-wiener theorem for reductivep-adic groups
复制标题
还原p-adic群的迹paley-wiener定理
DOI：
10.1007/bf02792538
复制
发表时间：
1986
期刊：
影响因子：
--
通讯作者：
D. Kazhdan
中科院分区：
文献类型：
--
作者：
J. Bernstein;Pierre Deligne;D. Kazhdan
定理1.1的陈述。设\(G\)是一个约化\(p\)-进群。群\(G\)在一个复向量空间\(E\)上的一个光滑表示\((\pi, E)\)被称为一个\(G\)-模。通常我们简化记号并写成\(\pi\)或者\(E\)。设\(\mathfrak{d}\pi(G)\)是\(G\)-模的范畴，\(\text{Irr }G\)是不可约\(G\)-模的等价类的集合，并且\(R(G)\)是有限长度的\(G\)-模的格罗滕迪克群；\(R(G)\)是一个以\(\text{Irr }G\)为基的自由阿贝尔群。我们固定一个极小抛物子群\(P_0\subset G\)以及它的列维分解\(P_0 = M_0\cdot U_0\)。通过一个标准列维子群我们指的是一个包含\(M_0\)的子群\(M\)，它是抛物子群\(P = M\cdot P_0\)（记号\(M\lt G\)）的一个列维分支。对于任何标准列维子群\(M\lt G\)，函子\(i_{G,M}:\mathfrak{d}\pi(M)\to\mathfrak{d}\pi(G)\)和\(r_{M,G}:\mathfrak{d}\pi(G)\to\mathfrak{d}\pi(M)\)定义了态射\(i_{G,M}:R(M)\to R(G)\)，\(r_{M,G}:R(G)\to R(M)\)（见[2,§]或者[1,2.5]）。设\(X_{\text{unr}}(G)\subset\{\chi:G\to\mathbb{C}^*\}\)是\(G\)的非分歧特征标群。它通过\(\chi:\pi\to\chi\pi\)自然地作用在\(\text{Irr }G\)和\(R(G)\)上。这个群有一个复代数群的自然结构（同构于\((\mathbb{C}^*)^d\)）。
1.2. 设\(\mathcal{H}(G)\)是\(G\)的赫克代数（\(G\)上具有紧支集的局部常值复值测度的代数）。每一个测度\(h\in\mathcal{H}(G)\)通过\(f_h(\pi)=\text{tr }\pi(h)\)定义了一个线性形式\(f_h:R(G)\to\mathbb{C}\)。很容易看到形式\(f = f_h\)满足以下条件：
(i) 对于任何标准列维子群\(M\lt G\)以及\(\sigma\in\text{Irr }M\)，函数\(\varphi\to f(i_{G,M}(\varphi\sigma))\)是复代数簇\(X_{\text{unr}}(M)\)上的一个正则函数。
(ii) 存在一个开紧子群\(K\subset G\)它控制\(f\)，即\(f\)仅在那些有非平凡的\(K\)-不变向量空间\(E^K\)的\(G\)-模\(E\)上非零。我们想要证明条件(i)-(ii)刻画了迹形式\(\{f_h\}\)。
w Statement of the theorem 1.1. Let G be a reductive p-adic group. A smooth representation (~', E) of the group G on a complex vector space E is called a G-module. Usually we shorten the notation and write w or E. Let d~(G) be the category of G-modules, Irr G the set of equivalence classes of irreducible G-modules, and R (G) the Grothendieck group of G-modules of fnite length; R(G) is a free abelian group with basis Irr G. We fix a minimal parabolic subgroup PoC G and its Levi decomposition P0 = Mo" Uo. By a standard Levi subgroup we mean a subgroup M _D Mo which is a Levi component of the parabolic subgroup P = M-Po (notation M < G). For any standard Levi subgroup M < G the functors iGu : d/t(M)-, d~(G) and rM~ : d~ (G)-~ d~ (M) define morphisms i~ : R (M)-~ R (G), rM~ : R (G)-~ R (M) (see [2, w or [1, 2.5]). Let xlt(G)C {t~: G-~C*} be the group of unramified characters of G. It acts naturally on Irr G and R(G) by if: 1r ~ ~Tr. This group has a natural structure of complex algebraic group (isomorphic to (C*)d). 1.2. Let ~(G) be the Hecke algebra of G (algebra of locally constant complex valued measures on G with compact support). Each measure h ~ ~(G) defines a linear form fh : R(G)-~C by/,(~-) = tr 7r(h). It is easy to see that the form f = fh satisfies the following conditions: (i) For any standard Levi subgroup M < G and or E IrrM, the function ~b ~ f(icM (~bo')) is a regular function on the complex algebraic variety xlt(M). (ii) There exists an open compact subgroup K C G which dominates f, i.e., f is nonzero only on the G-modules E, which have a nontriviai space E K of K-invariant vectors. We want to prove that the conditions (i)-(ii) characterize the trace forms {fh}.