The log-term of the Bergman kernel of the disc bundle over a homogeneous Hodge manifold

Abstract

We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kähler–Einstein manifold of classical type. Our results extends that in (Engliš and Zhang, Math Z 264(4):901–912, 2010) for the case of Hermitian symmetric spaces of compact type.

Appendix A: Classical Lie algebras

We now describe root systems, root vectors and Dynkin diagram for the classical groups $G = SU(d), Sp(d), SO(2d), SO(2d+1)$ following [17, Section III.8]. The diagrams are endowed with the equipment $\Pi _{\mathrm{can}}$ we have used throughout the paper and which we have referred to as the canonical equipment.

Example A.1

If $G= SU(d)$, $\mathfrak {g}^{{\mathbb C}} = sl(d, {\mathbb C})$ is the set of matrices with null trace, and let $\mathfrak {h}^{{\mathbb C}}$ be given by the diagonal matrices in $sl(d, {\mathbb C})$; for any $H = \mathrm {diag\;}(h_1, \ldots , h_d)$ let $e_i(H) = h_i$: then the root system is $R = \{ e_i - e_j \ | \ i \ne j \}$ and $E_{\alpha }$, $\alpha = e_i - e_j$, is the matrix $E_{ij}$ having 1 in the ij place and 0 anywhere else. The Killing form B satisfies $B(X,Y)= 2d \mathrm {tr}(XY)$ for all $X, Y \in sl(d, {\mathbb C})$. The canonical basis is

$$\begin{aligned} \Pi _{\mathrm{can}} = \{ \alpha _1 = e_1 - e_2, \ldots , \alpha _{d-1} = e_{d-1} - e_d \}. \end{aligned}$$

The Dynkin diagram, with this equipment, is

$$\begin{aligned} \underset{\begin{array}{c} \alpha _1 \end{array}}{\circ } - \underset{\begin{array}{c} \alpha _2 \end{array}}{\circ } - \cdots - \underset{\begin{array}{c} \alpha _{d-2} \end{array}}{\circ } - \underset{\begin{array}{c} \alpha _{d-1} \end{array}}{\circ } \\ \end{aligned}$$

Example A.2

For $G= Sp(d)$, $\mathfrak {g}^{{\mathbb C}} = sp(d, {\mathbb C})$ is the set of $2d \times 2d$ block matrices of the kind $\left( \begin{array}{ll} Z_1 &{} Z_2 \\ Z_3 &{} -{}^T Z_1 \\ \end{array} \right)$, where $Z_2, Z_3$ are symmetric. Let the Cartan subalgebra $\mathfrak {h}^{{\mathbb C}}$ be given by diagonal matrices $H = \mathrm {diag\;}(h_1, \ldots , h_d, -h_1, \ldots , -h_d)$ in $sp(d, {\mathbb C})$, and if for any such H we have $e_i(H) = h_i$, $i=1, \ldots , d$, then the root system is $R = \{ \pm e_i \pm e_j \}$ (the case $i=j$ is allowed when the signs are equal). The root vector $E_{\alpha }$ is given by $\left( \begin{array}{ll} E_{ij} &{} 0 \\ 0 &{} -E_{ji} \\ \end{array} \right)$ if $\alpha = e_i - e_j$, $\left( \begin{array}{cc} 0 &{} E_{ij} + E_{ji} \\ 0 &{} 0 \\ \end{array} \right)$ if $\alpha = e_i + e_j$ and $\left( \begin{array}{ll} 0 &{} 0 \\ E_{ij} + E_{ji} &{} 0 \\ \end{array} \right)$ if $\alpha = -e_i - e_j$ and the Killing form B is given by $B(X,Y)= 2(d+1) \mathrm {tr}(XY)$ for all $X, Y \in sp(d, {\mathbb C})$.

The canonical basis is given by

$$\begin{aligned} \Pi _{\mathrm{can}} = \{ \alpha _1 = e_1 - e_2, \ldots , \alpha _{d-1} = e_{d-1} - e_d, \alpha _d = 2 e_d \}. \end{aligned}$$

The Dynkin diagram, with this equipment, is

$$\begin{aligned} \underset{\begin{array}{c} \alpha _1 \end{array}}{\circ } - \underset{\begin{array}{c} \alpha _2 \end{array}}{\circ } - \cdots - \underset{\begin{array}{c} \alpha _{d-1} \end{array}}{\circ } \Leftarrow \underset{\begin{array}{c} \alpha _d \end{array}}{\circ } \end{aligned}$$

Example A.3

Let $G= SO(2d)$. Here and throughout the paper we identify the complexification $SO(2d, {\mathbb C})$ with the subgroup of $GL(2d, {\mathbb C})$ leaving invariant the quadratic form $z_1 z_{d+1} + \cdots + z_d z_{2d}$. Then $\mathfrak {g}^{{\mathbb C}} = so(2d, {\mathbb C})$ is the set of $2d \times 2d$ block matrices of the kind $\left( \begin{array}{cc} Z_1 &{} Z_2 \\ Z_3 &{} -{}^T Z_1 \\ \end{array} \right)$, where $Z_2, Z_3$ are skew-symmetric. Let the Cartan subalgebra $\mathfrak {h}^{{\mathbb C}}$ be given by diagonal matrices $H = \mathrm {diag\;}(h_1, \ldots , h_d, -h_1, \ldots , -h_d)$ in $so(2d, {\mathbb C})$, and if for any such H we have $e_i(H) = h_i$, $i=1, \ldots , d$, then the root system is $R = \{ \pm e_i \pm e_j \ (i \ne j) \}$. The root vector $E_{\alpha }$ is given by $\left( \begin{array}{ll} E_{ij} &{} 0 \\ 0 &{} -E_{ji} \\ \end{array} \right)$ if $\alpha = e_i - e_j$, $\left( \begin{array}{ll} 0 &{} E_{ij} - E_{ji} \\ 0 &{} 0 \\ \end{array} \right)$ if $\alpha = e_i + e_j$ ($i<j$) and $\left( \begin{array}{ll} 0 &{} 0 \\ E_{ij} - E_{ji} &{} 0 \\ \end{array} \right)$ if $\alpha = -e_i - e_j$ ($i<j$) and the Killing form B is given by $B(X, Y) =2(d-1)\mathrm {tr}(XY)$ for all $X ,Y \in so(2d, {\mathbb C})$.

The canonical basis is given by

$$\begin{aligned} \Pi _{\mathrm{can}} = \{ \alpha _1 = e_1 - e_2, \ldots , \alpha _{d-1} = e_{d-1} - e_d, \alpha _d = e_{d-1} + e_d \}. \end{aligned}$$

The Dynkin diagram, with this equipment, is

$$\begin{aligned} \underset{\begin{array}{c} \alpha _1 \end{array}}{\circ } - \underset{\begin{array}{c} \alpha _2 \end{array}}{\circ } - \cdots - \underset{\begin{array}{c} \alpha _{d-2} \end{array}}{\overset{\overset{\textstyle \circ _{\alpha _d}}{\textstyle \vert }}{\circ }} \,-\, \underset{\begin{array}{c} \alpha _{d-1} \end{array}}{\circ } \end{aligned}$$

Example A.4

Let $G= SO(2d+1)$. Here and throughout the paper we identify the complexification $SO(2d+1, {\mathbb C})$ with the subgroup of $GL(2d+1, {\mathbb C})$ leaving invariant the quadratic form $2(z_1 z_{d+1} + \cdots + z_d z_{2d}) + z_{2d+1}$. Then $\mathfrak {g}^{{\mathbb C}} = so(2d+1, {\mathbb C})$ is the set of $(2d+1) \times (2d+1)$ block matrices of the kind $\left( \begin{array}{lll} Z_1 &{} Z_2 &{} u \\ Z_3 &{} -{}^T Z_1 &{} v \\ -{}^T v &{} -{}^T u &{} 0 \end{array} \right)$, where $Z_2, Z_3$ are skew-symmetric and $u, v \in {\mathbb C}^d$. Let the Cartan subalgebra $\mathfrak {h}^{{\mathbb C}}$ be given by diagonal matrices $H = \mathrm {diag\;}(h_1, \ldots , h_d, -h_1, \ldots , -h_d, 0)$ in $so(2d+1, {\mathbb C})$, and if for any such H we have $e_i(H) = h_i$, $i=1, \ldots , d$, then the root system is $R = \{ \pm e_i \pm e_j \ (i \ne j), \pm e_i \}$. The root vector $E_{\alpha }$ is given by $\left( \begin{array}{lll} E_{ij} &{} 0 &{} 0 \\ 0 &{} -E_{ji} &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right)$ if $\alpha = e_i - e_j$, $\left( \begin{array}{lll} 0 &{} E_{ij} - E_{ji} &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right)$ if $\alpha = e_i + e_j$ ($i<j$), $\left( \begin{array}{lll} 0 &{} 0 &{} 0 \\ E_{ij} - E_{ji} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right)$ if $\alpha = -e_i - e_j$ ($i<j$), $\left( \begin{array}{lll} 0 &{} 0 &{} E_i \\ 0 &{} 0 &{} 0 \\ 0 &{} -{}^T E_i &{} 0 \end{array} \right)$ if $\alpha = e_i$ and $\left( \begin{array}{lll} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} E_i \\ -{}^T E_i &{} 0 &{} 0 \end{array} \right)$ if $\alpha = - e_i$ (being $E_i$ the ith vector of the canonical basis of ${\mathbb C}^d$) and the Killing form B is given by $B(X, Y) =(2d-1)\mathrm {tr}(XY)$ for all $X ,Y \in so(2d+1, {\mathbb C})$.

The canonical basis is given by

$$\begin{aligned} \Pi _{\mathrm{can}} = \{ \alpha _1 = e_1 - e_2, \ldots , \alpha _{d-1} = e_{d-1} - e_d, \alpha _d = e_d \}. \end{aligned}$$

The Dynkin diagram, with this equipment, is

$$\begin{aligned}&\underset{\begin{array}{c} \alpha _1 \end{array}}{\circ } - \underset{\begin{array}{c} \alpha _2 \end{array}}{\circ } - \cdots - \underset{\begin{array}{c} \alpha _{d-1} \end{array}}{\circ } \Rightarrow \underset{\begin{array}{c} \alpha _d \end{array}}{\circ } \end{aligned}$$

Remark A.5

In each of the above examples, the given Cartan subalgebra is the complexification of the Lie algebra of a maximal torus T in the compact group G (more precisely, for $G = SU(d), Sp(d), SO(2d), SO(2d+1)$ we have, respectively, $T = \mathrm {diag\;}(e^{i \theta _1}, \ldots , e^{i \theta _d}), \mathrm {diag\;}(e^{i \theta _1}, \ldots , e^{i \theta _d}, e^{-i \theta _1}, \ldots , e^{-i \theta _d})$, $\mathrm {diag\;}(e^{i \theta _1}, \ldots , e^{i \theta _d}, e^{-i \theta _1}, \ldots , e^{-i \theta _d})$, $\mathrm {diag\;}(e^{i \theta _1}, \ldots , e^{i \theta _d}, e^{-i \theta _1}, \ldots , e^{-i \theta _d}, 1)$). In general, given a maximal torus T in a compact connected Lie group G, its Lie algebra $\mathfrak {t}$ is a maximal abelian subalgebra of $\mathfrak {g}$ and its complexification $\mathfrak {t}^{{\mathbb C}}$ is a Cartan subalgebra of the complex Lie algebra $\mathfrak {g}^{{\mathbb C}}$.