#### Milestone Year

### 2012

#### Zhelobenko Invariants

At Weizmann, the normality of the commuting variety was settled
in 1997. This was a rather unexpected application of
representation theory to geometry. Indeed a key point was the
decomposition of the tensor product of two representations a
construction which plays a vital role in both nuclear and
elementary particle physics. Here we needed a delicate result in
this theory due to Kumar proving the Parthasarathy-Kostant conjecture.

In 2012 at Weizmann a long time open conjecture of Kostant for
the Clifford algebra of a semisimple Lie algebra was settled. It
used and invigorated the theory of Zhelobenko invariants in
representation theory. It gave a module for
Bernstein-Gelfand-Gelfand monoid introduced in their study of the
cohomology of the flag variety. Through Partharasathy-Ranga
Rao-Varadarajan determinants, an extended Harish-Chandra map was
shown to have as an image the space of Zhelobenko invariants,
both in the classical and quantum case. In the affine quantum
case, for which we showed in conjunction with an American
researcher sponsored by the Binational Science Foundation that
these determinants are infinite products, this question is still open.

Zhelobenko invariants were also known to play an important role
in the theory of Yangians via simple modules for "relative
Yangians". In 2012 at Weizmann the latter were classified by an
extension of a Bernstein-Gelfand equivalence of categories which
had been introduced to give an algebraic and more comprehensive
proof of Zhelobenko's classification of Harish-Chandra modules
for complex Lie groups.