A decorated surface ${\mathbb{S}}$ is an oriented topological surface with marked points on the boundary considered modulo the isotopy. We consider the moduli space ${\mathcal{M}}_{{\mathbb{S}}}$ of hyperbolic structures on ${\mathbb{S}}$ with geodesic boundary, such that the hyperbolic structure near each marked point is a cusp, equipped with a horocycle. The space ${\mathcal{M}}_{{\mathbb{S}}}$ carries a volume form $\Omega $ . Let us fix the set ${{\mathrm{K}}}$ of the distances between the horocycles at the adjacent cusps, and the set ${\mathrm{L}}$ of the geodesic lengths of boundary circles without cusps. We get a subspace ${\mathcal{M}}_{{\mathbb{S}}}({{\mathrm{K}}}, {\mathrm{L}})$ with the induced volume form $\Omega _{{\mathbb{S}}}({{{\mathrm{K}}}, {\mathrm{L}}})$ . However, if the cusps are present, the volume of the space ${\mathcal{M}}_{{\mathbb{S}}}({{\mathrm{K}}}, {\mathrm{L}})$ , or its variant without horocycles, is infinite. We introduce the exponential volume form $e^{-{W}}\Omega $ , where ${W}$ is the potential - a positive function on ${\mathcal{M}}_{{\mathbb{S}}}$ , given by the sum over the cusps of the hyperbolic areas under the horocycles. We show that the following exponential volume is finite: 1 \( \int _{{\mathcal{M}}_{{\mathbb{S}}}({{\mathrm{K}}}, {\mathrm{L}})}e^{-{W}}\Omega _{{ \mathbb{S}}}({{{\mathrm{K}}}, {\mathrm{L}}}). \) We suggest that moduli spaces ${\mathcal{M}}_{{\mathbb{S}}}({{\mathrm{K}}}, {\mathrm{L}})$ with the exponential volume forms are the true analogs of moduli spaces ${\mathcal{M}}_{g,n}$ with the Weil–Petersson volume form, e.g. relevant to the open string theory. We prove unfolding formulas, expressing integrals $\int _{{\mathcal{M}}_{{\mathbb{S}}}({{\mathrm{K}}}, {\mathrm{L}})}f\ e^{-{W}}\Omega _{{ \mathbb{S}}}({{{\mathrm{K}}}, {\mathrm{L}}})$ , where $f$ is an integrable function, as finite sums of similar integrals for the three elementary decorated surfaces. They generalise Mirzakhani’s recursions for the volumes of moduli spaces of hyperbolic surfaces. We show that exponential volumes for elementary decorated surfaces give rise to a commutative algebra ℰ, which we call the positive Hecke-Whittaker algebra for ${\mathrm{PGL}}_{2}({\mathbb{R}})$ . Exponential volumes for all decorated surfaces and unfolding formulas extend the algebra ℰ to all decorated surfaces.
