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.