THE HIDDEN SEMI-INVARIANTS GENERATORS OF AN ALMOST-FROBENIUS BIPARABOLIC

Abstract

Let $\mathfrak{a}$ be an algebraic Lie algebra and A its adjoint group. One calls $\mathfrak{a}$ Frobenius if A has a dense orbit in $\mathfrak{a}$ ^*. In this case the algebra Sy($\mathfrak{a}$) spanned by the A semi-invariant polynomials on $\mathfrak{a}$ ^* is not reduced to scalars and is polynomial. Again d := dim a is even and after Ooms [27] the generators all appear as factors of the coe_cient of ^^d $\mathfrak{a}$ in the d = 2-th power of the Poisson bivector of $\mathfrak{a}$.

Though appealing, this result does not give the weights of the generators, nor their degrees. In the present work we compute these weights and degrees in the case when $\mathfrak{a}$ is a Frobenius biparabolic subalgebra $\mathfrak{q}$ of a simple Lie algebra $\mathfrak{g}$. These results are relatively easy consequences of our earlier work on biparabolic subalgebras [7], [16], [18], except for the question of the existence of \hidden generators". The latter have squares which can be obtained by a general construction using the Hopf dual U($\mathfrak{q}$)^★ of U($\mathfrak{q}$), yet are themselves not easily proven to exist. These \hidden generators" only occur outside types A and C.

To construct these hidden generators, the notion of a more general almost-Frobenius biparabolic subalgebra is introduced. It is shown that Sy($\mathfrak{a}$) is polynomial if q is an almost-Frobenius biparabolic and may itself have \hidden generators" outside types A and C. However, it is possible to construct these hidden generators through a rather precise description of the generators of an almost-Frobenius biparabolic subalgebra of a simple Lie algebra of type A.