ON THE REGULARITY OF D -MODULES GENERATED BY RELATIVE CHARACTERS
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Springer Science+Business Media New York (2020)
Transformation Groups
ON THE REGULARITY OF D-MODULES GENERATED BY RELATIVE CHARACTERS WEN-WEI LI∗ Beijing International Center for Mathematical Research / School of Mathematical Sciences Peking University No. 5 Yiheyuan Road Beijing 100871 People’s Republic of China [email protected]
Abstract. Following the ideas of Ginzburg, for a subgroup K of a connected reductive Rgroup G we introduce the notion of K-admissible D-modules on a homogeneous G-variety Z. We show that K-admissible D-modules are regular holonomic when K and Z are absolutely spherical. This framework includes: (i) the relative characters attached to two spherical subgroups H1 and H2 , provided that the twisting character χi factors through the maximal reductive quotient of Hi , for i = 1, 2; (ii) localization on Z of Harish-Chandra modules; (iii) the generalized matrix coefficients when K(R) is maximal compact. This complements the holonomicity proven by Aizenbud–Gourevitch–Minchenko. The use of regularity is illustrated by a crude estimate on the growth of K-admissible distributions based on tools from subanalytic geometry.
Contents 1. Introduction 2. Equivariant and monodromic D-modules 3. k-admissible D-modules: holonomicity 4. Review of horocycle correspondence 5. K-admissible D-modules: regularity 6. Subanalytic sets and maps 7. Growth conditions 8. Growth of regular holonomic solutions 9. Applications to admissible distributions 10. The case of generalized matrix coefficients DOI: 10.1007/S00031-020-09624-x Supported by NSFC-11922101. Received July 23, 2019. Accepted July 15, 2020. Corresponding Author: Wen-Wei Li, e-mail: [email protected] ∗
WEN-WEI LI
1. Introduction Let G be a connected reductive group over R and denote its opposite group by Gop . Differential equations with regular singularities have played an important role in representation theory of the Lie group G(R). One significant example is Harish-Chandra’s study of invariant eigendistributions on G(R), which includes the character f 7→ Θπ (f ) := tr π(f ),
f ∈ Cc∞ (G(R))
of an SAF representation π of G(R) as a typical case. Our terminology of SAF representation follows [6], meaning smooth admissible Fr´echet of moderate growth. Another example is the study of asymptotics of the matrix coefficients hˇ v , π(·)vi ∈ C ∞ (G(R)),
v ∈ Vπ , vˇ ∈ Vπˇ
of these representations, as exemplified by [11]; here Vπ stands for the underlying Fr´echet space, π ˇ for the contragredient representation, and h· , ·i for the canonical pairing. Generalizing the characters or matrix coefficients to the relative setting, one can also consider similar distributions on Z(R), where Z is an R-variety with Z(R) 6= ∅, homogeneous under right G-action, satisfying finiteness condition under some subgroup K ⊂ G and the center Z(g) of U (g). Of course, Z and K must be subject to some geometric conditions. It turns out that sphericity is a reasonable requirement. In this article, we say Z is spherical if ZC := Z ×R C has an open dense orbit under any Bo
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