On Gustafson integrals for the group $\mathrm{SL}(2,\mathbb{R})$

In this paper, we calculate the Gustafson integrals of the first and second types for the group $\mathrm{SL}(2,\mathbb{R})$ in the case of an integration domain of a special form. The definitions of the analogs of the sine, cosine and Gamma functions are given, and their main properties are formulated. The conclusion lists open questions.