Steinberg modules and high-dimensional rational cohomology of symplectic groups

Robin J. Sroka (University of Münster)

Wednesday 8th November, 2023 16:00-17:00 Maths 110

Abstract

While Borel completely computed the rational cohomology of Sp_{2n}(Z) in degrees that are small compared to n, it remains largely mysterious in high degrees. In this talk, I will discuss what is known and present recent work exploring these high-dimensional cohomology groups. The starting point is a duality theorem of Borel-Serre, which shows that the rational cohomology of Sp_{2n}(Z) is trivial in degrees greater than n^2 and that, in degree n^2 and below, it is controlled by a complicated Sp_{2n}(Z)-representation: the symplectic Steinberg module. I will explain how a suitable generating set and presentation of this module can be used to calculate the rational cohomology of Sp_{2n}(Z) in degree n^2 and n^2-1 and discuss connections to the study of moduli spaces, a conjecture of Church-Farb-Putman as well as guiding analogies with other arithmetic groups such as SL_{n}(Z). This talk is primarily based on joint works with Benjamin Brück and Peter Patzt.

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