The paper considers the heat equation associated to a multiplicity function on a root system and describes the image of the corresponding Segal-Bargmann transform on a space of holomorphic functions. First let us recall some material for “the geometric case”, i.e. for a Riemannian symmetric space \(G/K\) of non-compact type. Here \(G\) is a connected semisimple Lie group with finite center, \(K\) a maximal compact subgroup of \(G\). Let \(\mathfrak g\) and \(\mathfrak k\) be the Lie algebras of \(G\) and \(K\) respectively. There is the canonical decomposition \(\mathfrak g= \mathfrak k + \mathfrak p\). Let \(\mathfrak a\) be a maximal compact subspace of \(\mathfrak p\) and \(\Delta\) the root system of the pair \((\mathfrak g, \mathfrak a)\). Let \(\Delta^+\) be a positive system and \(\rho\) be half the sum of positive roots with multiplicities: \(\rho=(1/2)\sum m_\alpha \, \alpha\). Denote \(A=\text{ exp} \, \mathfrak a\) (\(A\cong \mathfrak a\)). The radial part \(L\) of the Laplace-Beltrami operator on \(G/K\) is the following differential operator on \(A\):
\[ L=L_A+\sum_{\alpha\in\Delta^+} \, m_\alpha \, \text{ coth} \, \alpha\circ h_\alpha \]
where \(L_A\) is the usual Laplace-Beltrami operator on \(A\), the elements \(h_\alpha\) are defined by \((h_\alpha,X)=\alpha(X)\) for all \(X\in\mathfrak a\), the inner product \((X,Y)\) is the Killing form. In this formula the vectors \(h_\alpha\) are considered as differential operators. Harmonic analysis on \(G/K\) is reduced to harmonic analysis of the functions in \(L^2(G/K)^K\). This space is isomorphic to the space \(L^2(A,d\mu(a))^W\) where \(W\) is the Weyl group of the pair \((\mathfrak g, \mathfrak a)\), the measure \(d\mu(a)\) is \(\delta(a)da\) with the density function \(\delta(a)= \prod | 2 \, \text{ sinh} \, \alpha(\text{ log} \, a)| ^{m_\alpha}\), the product is taken over \(\alpha\in \Delta^+\). The Fourier transform \(\mathcal F\) assigns to a function \(f(a)\) in \(L^2(A,d\mu(a))^W\) the following function \(F(\lambda)\) on \({\mathfrak a}^*\):
\[ F(\lambda)=\int_A \, f(a) \, \varphi_{-i\lambda}(a) \, d\mu(a) \]
where \(\varphi_{\mu}(x)\), \(\mu\in{\mathfrak a}^*_{\mathbb C}\), is a spherical function on \(G/K\). The function \(f(a)\) is recovered by its Fourier transform:
\[ f(a)=\frac{1}{| W| ^2} \int_{\alpha^*} \, F(\lambda) \, \varphi_{i\lambda}(a) \, d\nu(\lambda) \tag{1} \]
where \(d\nu(\lambda)=| c(i\lambda)| ^{-2}d\lambda\) and \(c(\mu)\) is the Harish-Chandra \(c\)-function, an explicit formula for it is given by Gindikin and Karpelevich. Consider the Cauchy problem for the heat equation on \(G/K\). Assume that the initial function is invariant with respect to \(K\). Then the solution is also invariant with respect to \(K\). Thus, the problem is reduced to the Cauchy problem for \(A\):
\[ {\partial}_t u(a,t)=Lu(a,t), \;\;u(a,t)| _{t=0}=f(a). \tag{2} \]
Now the problem can be generalized: we forget the space \(G/K\) itself and keep the Lie algebra \(\mathfrak a\) with a root system \(\Delta\) and the space \(A\), and we replace the dimensions \(m_\alpha\) by an arbitrary \(W\)-invariant function \(m_\alpha\geqslant 0\) of \(\alpha\in\Delta\) (a multiplicity function). Then we obtain generalizations of the above-mentioned objects: \(L\), \(\delta(a)\), \(\varphi_\mu(a)\), \(\mathcal F\) etc., they depend on this function \(m_\alpha\). In particular, \(\varphi_\mu(a)\) is the Heckman-Opdam hypergeometric function, \(\mathcal F\) is called the hypergeometric Fourier transform etc. The solution \(u(a,t)\) of the problem (2) with \(f\in L^2(A,d\mu(a))^W\) for an arbitrary multiplicity function is given by the right hand side of (1) with \(F(\lambda)\) multiplied by \(\text{ exp}\{-t(| \lambda| ^2+| \rho| ^2)\}\). This solution can be extended to a \(W\)-invariant function \(u(z,t)\) on \(A \, \text{ exp} \, i\Omega\), where \(\Omega\) is the set of \(X\in\mathfrak a\) such that \(| \alpha(X)| <\pi/2\) for all \(\alpha\in\Delta\) (now ”exp” means the factorization over a lattice). The map \(H_t\) assigning to a function \(f(a)\) the function \(u(z,t)\) is called the Segal-Bargmann transform associated with the hypergeometric heat equation. The main result of the paper consists of a description of the image \({\mathcal H}_t\) of \(H_t\), the space \({\mathcal H}_t\) is endowed with an inner product so that \({\mathcal H}_t\) becomes a Hilbert space and the map \(H_t: L^2(A,d\mu(a))^W \rightarrow {\mathcal H}_t\) is a unitary isomorphism. The main idea is to reduce the problem to the classical case (\(m_\alpha=0\)) – by means of the operator \(\Lambda=| W| ^{-1}{\mathcal F}_A^{-1}\circ\Psi\circ{\mathcal F}\) where \({\mathcal F}_A\) is a usual Fourier transform and \(\Psi\) is the multiplication by \(c(-i\lambda)^{-1}\). The operator \(\Lambda\) has an important property: it carries the operator \(L\) to the operator \(L_A-| \rho| ^2\), i.e. \(\Lambda\circ L=(L_A-| \rho| ^2)\circ\Lambda\), and it is a unitary isomorphism \(L^2(A,d\mu(a))^W \rightarrow L^2(A,da)^{\tau(W)}\). Here \(\tau\) is a representation of \(W\), it acts on \(L^2({\mathfrak a}^*,d\lambda)\) by \((\tau(w)F)(\lambda)=\{c(iw\lambda)/c(i\lambda)\} F(w^{-1}\lambda)\) and therefore on \(L^2(A,da)\). In the case \(m_\alpha=2\) (it corresponds to \(G/K\) with \(G\) complex) the results of the paper give a result of B. Hall and J. Mitchell [J. Funct. Anal. 227, No. 2, 338–371 (2005; Zbl 1082.58038)].