Canonical bases in representation theory

Since the very beginning, the role of bases in simple modules over finite and Lie groups was critical: at the very least they were indispensable in computing dimensions and characters. This role was played well by Young tableaux bases in Specht modules for symmetric groups S_n and Gelfand-Tsetlin bases for simple GL_n(C)-modules, to name a few. 

After the groundbreaking Kazhdan-Lusztig paper of 1979, it became clear that in order to obtain an interesting basis in each simple module of a group algebra of a finite group, one has to first deform it and then construct a (canonical) basis in the so deformed algebra.  This canonical basis projects to each simple module and becomes a basis there. 

A similar idea on the Lie-theoretic side was coined by Gelfand and Zelevinsky in 1983: to find bases in all simple modules V_lambda over GL_n(C) one can naturally realize V_lambda as a subspace of the coordinate algebra C[N] of the group N of unipotent nxn matrices and find a  good basis there which will naturally descend to each V_lambda under the embedding into C[N], and will eventually solve the tensor product multiplicity problem (for reductive or semisimple Lie groups G with the maximal unipotent subgroup N they expected same outcome). 

With the discovery of quantum groups in 1986, this idea was implemented in a very surprising (and in a sense dual way) way by George Lusztig in 1989 who noticed that the quantized enveloping algebra U_q(nn), where nn is the Lie algebra  of N, admits a canonical basis B which, on the one hand, was constructed in Kazhdan-Lusztig fashion (by means of the celebrated "Lusztig's Lemma"), and on the other hand, descends to each simple module V_lambda over the full quantum group U_q(gg), where gg is the Lie algebra of G (shortly after Lusztig's amazing discovery, Masaki Kashiwara in 1990 recovered Lusztig's constructions via the theory of crystal bases which emerge when q=0). 

In our work with Andrei Zelevinsky in 1993 we adapted a dual approach by taking advantage of the fact that U_q(nn) is not only a q-deformation of the universal enveloping algebra U(nn), but also is a q-deformation of its Hopf dual C[N], thus claiming that the dual basis B^dual of B fits Gelfand-Zelevinsky framework (our approach had an unexpected byproduct:  the modern theory of cluster algebras emerged from the study of B^dual for various semisimple or even even Kac-Moody Lie groups).

In the second part of the talk, I will reveal that the aforementioned dual canonical bases of U_q(nn) and of its remarkable subalgeras U_q(w), known as quantum Schubert cells, are also obtained by the application of the (very innocently looking!) Lusztig Lemma and if time permit, I will demonstrate how to obtain a canonical basis in quantum Heisenberg algebra H_q(gg) and, ultimately, in U_q(gg) along these lines. This is joint work with Jacob Greenstein, 2014-2017.