Graded characters of induced representations for real reductive Lie groups. I
Induced Harish-Chandra modules or modules in category \({\mathfrak o}'\) are studied in terms of corresponding mixed perverse sheaves equipped with Frobenius actions. This correspondence is used to compute the socle and cosocle filtrations in several situations. Indeed, in these instances there is a correspondence between socle or cosocle filtrations with the weight filtration of the corresponding sheaf. Using the weight filtration, an invariant, the graded character, is attached to these modules. Two induced modules having the same graded character are said to be weakly isomorphic. The weak isomorphism classes of principal series modules for R-split groups are shown to coincide with the isomorphism classes for the modules. Moreover a conjecture extending this to all principal series modules is discussed.
scientific article; zbMATH DE number 4127525
scientific article