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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.00370 |
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| _version_ | 1866917300587724800 |
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| author | Shimada, Masafumi |
| author_facet | Shimada, Masafumi |
| contents | Let $G$ be a connected semisimple Lie group, and $G_0$ be its connected split real form. In this paper, we deduce explicit expressions for the heat kernels $ρ^{G_0}_t$ associated with the Laplace--Beltrami operators $Δ_{G_0}$ and $Δ_{G}$ respectively, using the algebra of differential operators on an appropriate homogeneous space. These expressions involve the heat Gaussian and the heat kernel on a maximal compact subgroup. Using these expressions for $ρ^{G_0}_t$ and $ρ^{G}_t$, we derive an integral formula relating the heat kernel $ρ^{G_0}_t$ to $ρ^{G}_t$. In the special case of $G_0=SL(2,\mathbb{R})$, we show that the integral formula of $ρ^{SL(2,\mathbb{R})}$ is expressed in terms of the properties of Tchebycheff polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00370 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The heat kernel on a complex semisimple Lie group and an integral presentation of the heat kernel on its split real form Shimada, Masafumi Functional Analysis 22E30, 43A80, Let $G$ be a connected semisimple Lie group, and $G_0$ be its connected split real form. In this paper, we deduce explicit expressions for the heat kernels $ρ^{G_0}_t$ associated with the Laplace--Beltrami operators $Δ_{G_0}$ and $Δ_{G}$ respectively, using the algebra of differential operators on an appropriate homogeneous space. These expressions involve the heat Gaussian and the heat kernel on a maximal compact subgroup. Using these expressions for $ρ^{G_0}_t$ and $ρ^{G}_t$, we derive an integral formula relating the heat kernel $ρ^{G_0}_t$ to $ρ^{G}_t$. In the special case of $G_0=SL(2,\mathbb{R})$, we show that the integral formula of $ρ^{SL(2,\mathbb{R})}$ is expressed in terms of the properties of Tchebycheff polynomials. |
| title | The heat kernel on a complex semisimple Lie group and an integral presentation of the heat kernel on its split real form |
| topic | Functional Analysis 22E30, 43A80, |
| url | https://arxiv.org/abs/2603.00370 |