Holomorphic functions and subelliptic heat kernels over Lie groups

Abstract

Abstract. A Hermitian form q on the dual space, g * , of the Lie algebra, g, of a Lie group, G, determines a sub-Laplacian, , on G. It will be shown that Hörmander's condition for hypoellipticity of the sub-Laplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universal enveloping algebra, U , is non-degenerate. The subelliptic heat semigroup, e t /4 , is given by convolution by a C ∞ probability density ρ t . When G is complex and u : G → C is a holomorphic function, the collection of derivatives of u at the identity in G gives rise to an element, u(e) ∈ U . We will show that if G is complex, connected, and simply connected then the "Taylor" map, u →û(e), defines a unitary map from the space of holomorphic functions in L 2 (G, ρ t ) onto a natural Hilbert space lying in U .

Surjectivity of the Taylor map for complex nilpotent Lie groups

“…In [6] we have shown that indeed such a unitarity theorem holds if the subLaplacian is merely subelliptic. The inner product on g must, properly speaking, be defined on the dual space g M in this case and is allowed to be degenerate to the extent that Hörmander's theorem for hypoellipticity of the subLaplacian permits.…”

“…2. By Hörmander's theorem [16], Lie(H) = g iff is hypoelliptic, see the end of Section 1 in [6] for a more detailed discussion on this last point. So under Hörmander's condition on q, the operator, , in (2·9) admits a smooth heat kernel,…”

“…In Section 3 we will give a proof of surjectivity in this case. It is a very different proof from the general surjectivity proof in [6]. Moreover, along the way it produces a proof that the finite rank tensors are dense in the tensor-side Hilbert space if G is graded nilpotent and the Laplacian is adapted to the gradation.…”

“…It can be shown that and (H, (·, ·) H ) only depend on q and not on the choice of {X j } m j=1 ⊂ g for which (2·8) holds, see [6].…”

“…In this section we will review some notation and basic results from [6]. We will use angle brackets, ·, · , to denote the pairing of a vector space, V, and its algebraic dual, V , i.e.…”