preprint Igor Burban author 72064 Yuriy Drozd author In this article we study the principal block of the category of real Harish-Chandra modules for the group $\mathsf{SL}_2(\RR)$ and relate it to the category of finite dimensional modules over the so-called real Gelfand order. We describe several distinguished classes of the corresponding indecomposable representations. 2026 eng 2605.18000 Burban, Igor, and Yuriy Drozd. “Representation Theory of the Real Gelfand Order and Real Harish-Chandra Modules for SL_2(R).” <i>ArXiv:2605.18000</i>, 2026. Burban I, Drozd Y. Representation theory of the real Gelfand order and real Harish-Chandra modules for SL_2(R). <i>arXiv:260518000</i>. Published online 2026. @article{Burban_Drozd_2026, title={Representation theory of the real Gelfand order and real Harish-Chandra modules for SL_2(R)}, journal={arXiv:2605.18000}, author={Burban, Igor and Drozd, Yuriy}, year={2026} } Burban, I., &#38; Drozd, Y. (2026). Representation theory of the real Gelfand order and real Harish-Chandra modules for SL_2(R). In <i>arXiv:2605.18000</i>. I. Burban and Y. Drozd, “Representation theory of the real Gelfand order and real Harish-Chandra modules for SL_2(R),” <i>arXiv:2605.18000</i>. 2026. Burban, Igor, and Yuriy Drozd. “Representation Theory of the Real Gelfand Order and Real Harish-Chandra Modules for SL_2(R).” <i>ArXiv:2605.18000</i>, 2026. I. Burban, Y. Drozd, ArXiv:2605.18000 (2026). 662922026-07-07T06:22:32Z2026-07-07T06:23:43Z