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Please join us for pre-seminar Tea

Consider a semi-simple Lie group, $G$ and a discrete torsion-free subgroup $\Gamma < G$. When $G$ has rank one, there is an important, well-understood connection between the growth rate of elements of $\Gamma$-orbits, bounds on the spectrum, and temperedness, or more generally properties of the unitary $L^2$ representation (due, in part, to work of Elstrodt and Patterson). In this talk I will present an extension of this connection to higher rank.  This is joint work with Tobias Weich and Lasse Wolf. 

Classifying the irreducible unitary representations of a real reductive Lie group is one of the oldest unsolved problems in representation theory. In this talk, I will discuss two conceptual approaches to this problem, Hodge theory and the orbit method, and the relationship between them. The Hodge-theoretic approach, proposed by Schmid and Vilonen in 2011, exploits links between representation theory and the topology of algebraic varieties to endow irreducible representations with canonical Hodge filtrations. In joint work with Vilonen, we have shown that these filtrations detect unitary representations. The orbit method, on the other hand, is an old idea (dating back to the 1960s) that unitary representations should arise naturally as quantisations of certain classical spaces, the co-adjoint orbits. In joint work in preparation with Lucas Mason-Brown, we show that the Hodge filtration realises this expectation for a key class of representations, called unipotent, and deduce that these representations are always unitary.