# Algebra Seminar

This seminar is held on Tuesdays at 4pm in 210 University Hall.

### Fall Quarter, 2021

- October 5, Organizational Meeting
- October 12,
**Ben Elias** (UO)

Quantum geometric Satake and K-theory
**Abstract**: There’s a certain geometric setup where, when you apply the equivariant cohomology functor, you get something called singular Soergel bimodules. The geometric Satake equivalence states that these singular Soergel bimodules are equivalent to representations of a semisimple lie algebra. I’ll explain this in easy examples – geometric Satake for dummies, if you will.

If instead you apply the equivariant K-theory functor, you get K-theoretic singular Soergel bimodules, a new concept due to myself and Geordie Williamson. Our quantum geometric Satake equivalence (conjecture) roughly states that K-theoretic singular Soergel bimodules are equivalent to representations of the corresponding quantum group. As stated, this is false, and the fix is rather surprising.

- October 26,
**Alexandra Utiralova** (MIT)

Harish-Chandra bimodules in complex rank
**Abstract**: Deligne tensor categories are defined as an interpolation of the categories of representations of groups GL_n, O_n, Sp_{2n} or S_n to the complex values of the parameter n. One can extend many classical representation-theoretic notions and constructions to this context. These complex rank analogues of classical objects provide insights on their stable behavior patterns as n goes to infinity.

I will talk about some of my results on Harish-Chandra bimodules in Deligne categories. It is known that in the classical case simple Harish-Chandra bimodules admit a classification in terms of W-orbits of certain pairs of weights. However, the notion of weight is not well-defined in the setting of Deligne categories. I will explain how in complex rank the above-mentioned classification translates to a condition on the corresponding (left and right) central characters.

Another interesting phenomenon arising in complex rank is that there are two ways to define Harish-Chandra bimodules. That is, one can either require that the center acts locally finitely on a bimodule M or that M has a finite K-type. The two conditions are known to be equivalent for a semi-simple Lie algebra in the classical setting, however, in Deligne categories that is no longer the case. I will talk about a way to construct examples of Harish-Chandra bimodules of finite K-type using the ultraproduct realization of Deligne categories.

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