The Complex-time Segal–Bargmann transform

Abstract We introduce a new form of the Segal–Bargmann transform for a Lie group K of compact type. We show that the heat kernel ( ρ t ( x ) ) t > 0 , x ∈ K has a space-time analytic continuation to a holomorphic function ( ρ C ( τ , z ) ) Re τ > 0 , z ∈ K C , where K C is the complexification of K. The new transform is defined by the integral ( B τ f ) ( z ) = ∫ K ρ C ( τ , z k − 1 ) f ( k ) d k , z ∈ K C . If s > 0 and τ ∈ D ( s , s ) (the disk of radius s centered at s), this integral defines a holomorphic function on K C for each f ∈ L 2 ( K , ρ s ) . We construct a heat kernel density μ s , τ on K C such that, for all s , τ as above, B s , τ : = B τ | L 2 ( K , ρ s ) is an isometric isomorphism from L 2 ( K , ρ s ) onto the space of holomorphic functions in L 2 ( K C , μ s , τ ) . When τ = t = s , the transform B t , t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When τ = t ∈ ( 0 , 2 s ) , the transform B s , t coincides with the one introduced by the first two authors.