We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the ordinary restriction of a character sheaf on the compactification to a semi-stable stratum is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16

We present a series of results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves, including a simple formula of the <i>n</i>-point functions for Witten9s classes, an effective recursion formula to compute higher WeilPetersson volumes, several new recursion formulae of intersection numbers and our proof of a conjecture of Itzykson and Zuber concerning denominators of intersection numbers. We also present Virasoro and KdV properties of generating functions of general mixed and intersections.

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Tautological systems are Picard-Fuchs type systems arising from varieties with large symmetry. In this survey, we discuss recent progress on the study of tautological systems. This includes tautological systems for vector bundles, a new construction of Jacobian rings for homogenous vector bundles, and relations between period integrals and zeta functions.

Tautological systems developed in [8, 9] are Picard-Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to zero loci of global sections of vector bundles. In particular, we obtain similar criterion as in [8, 9] about holonomicity and regularity of the systems. We also prove solution rank formulas and geometric realizations of solutions following the work on hypersurfaces in homogeneous varieties [4].