Abstract:

It has been understood for some time now that many highlights of Lie
theory, such as the representation-theoretic theory of special
functions, or the Kazhdan–Lusztig theory, have a natural extension to a
much broader setting, the boundaries of which are yet to be explored. In
this extension, the focus is shifting from a group G to various classes
of algebraic varieties that possess the key features of T*G/B. While
there are some proposal about what should replace a Lie algebra, root
systems, etc., it is less clear what should be the group, or
multiplicative analog of these structures. Reflecting the nature of the
field, the talk will combine a review of established partial results
with unsubstantiated speculations.