I. M. GEL'FAND. - The cohomology of infinite dimensional Lie algebras; some questions of integral geometry · 1970-lie-163

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The exact assertion is unresolved. Gel'fand posed it for the fixed weight $e^{-H}$ and reported that it had been checked for $SU(3)$, in addition to the classical $SU(2)$ Bargmann case, but gave no general proof. I found no later source proving closure under the adjoint defined by this particular weight for every simply connected compact semisimple $K$, nor a counterexample.

The closest detailed follow-up changes an essential datum. In Differential Operators on the Base Affine Space and a Study of $\mathfrak g$-Modules, I. N. Bernstein, I. M. Gel'fand, and S. I. Gel'fand, in Lie Groups and Their Representations (Proceedings of the 1971 Bolyai Janos Summer School), Halsted Press/Akademiai Kiado, 1975, pp. 21-64 (direct PDF), Section 5 replaces $e^{-H}$ by an unknown weight $u(\sum_i q_iH_i)$, states adjoint-closure as Conjecture II, proves the $SL(2,\mathbb C)$ case, and verifies $SL(3,\mathbb C)$ only for a specially chosen $u$ satisfying an ODE. Thus it does not settle the fixed $u(t)=e^{-t}$ problem stated here.

Substantial algebraic progress on the same operator ring does not close this analytic gap:

• N. N. Shapovalov, Structure of the Algebra of Regular Differential Operators on a Basic Affine Space, Functional Analysis and Its Applications 8 (1974), 37-46, doi:10.1007/BF02028306, studies the algebraic structure of $\mathscr D(G/U)$, but not the adjoint for Gel'fand's weighted holomorphic Hilbert space.
• Roman Bezrukavnikov, Alexander Braverman, and Leonid Positselskii, Gluing of Abelian Categories and Differential Operators on the Basic Affine Space, Journal of the Institute of Mathematics of Jussieu 1 (2002), 543-557, doi:10.1017/S1474748002000154, constructs the Weyl-group action on $\mathscr D(G/U)$ using partial Fourier transforms and proves Noetherian and homological properties. Its unitary Fourier transforms act on a different $L^2$-space attached to a split real form, not on the $e^{-H}$ holomorphic space in the problem.
• Thierry Levasseur and J. T. Stafford, Differential Operators and Cohomology Groups on the Basic Affine Space, in Studies in Lie Theory, Progress in Mathematics 243, Birkhauser, 2006, pp. 377-403, doi:10.1007/0-8176-4478-4_14, Theorem 3.6 gives a finite generating description of $\mathscr D(G/U)$ by Weyl-transformed regular functions and Cartan vector fields. It does not identify the adjoints of those generators for the fixed weight $e^{-H}$.
• Victor Ginzburg and David Kazhdan, Differential Operators on $G/U$ and the Gelfand-Graev Action, Advances in Mathematics 403 (2022), 108368, doi:10.1016/j.aim.2022.108368, gives an algebraic construction of the Gelfand-Graev Weyl-group action, again without treating the weighted Hilbert-space adjoint in the statement.

The original source is I. M. Gel'fand, The Cohomology of Infinite Dimensional Lie Algebras; Some Questions of Integral Geometry, in Actes du Congres International des Mathematiciens, Nice, 1970, vol. 1, Gauthier-Villars, 1971, pp. 95-111 (direct proceedings PDF). It explicitly formulates the fixed-weight conjecture and reports the $SU(3)$ check. Beyond these low-rank reports and results for modified weights, the all-groups fixed-weight assertion remains the unfilled part.