Extending tensors on polar manifolds

Abstract

Given a polar action on a Riemannian manifold, we prove surjectivity of restriction to the section for general invariant tensors, and a sharper surjectivity result in the special case of metrics. These are related to the Chevalley Restriction Theorem and Michor’s Basic Forms Theorem. The proofs rely on results in the Invariant Theory of finite reflection groups and symmetric pairs, some of which may be of independent interest.

Appendix: Hessian Theorem for finite reflection groups

In this section we provide proofs of Theorem 4 and Observation 2. The latter relies on Calculation 1, which can be checked with a computer. See [22] for the source code for a script written in GAP [26] using the package CHEVIE [12], which performs these calculations.

Note that as far as the proofs of Theorem 2 and Observation 1 are concerned, one only needs to consider crystallographic reflection groups (see [14] for a definition). Our proofs include the non-crystallographic cases for the sake of completeness.

Recall some facts about finite reflection groups: First, the algebra of invariants, $A=\mathbb {R}[\varSigma ]^W$, is a free polynomial algebra with n generators, where $n=\dim \varSigma$. This is known as Chevalley’s Theorem—see [1, Chapter V]. Such a set $\{\rho _i\}$ of homogeneous generators is called a set of basic invariants, and $d_i=\deg \rho _i$ are called the degrees of W.

Second, $\mathbb {R}[\varSigma ]$ is a free $A=\mathbb {R}[\varSigma ]^W$-module, more precisely $\mathbb {R}[\varSigma ]=A\otimes H$, where H is isomorphic to the regular representation of W (see Theorem B in [3]). In particular, ${\mathcal {M}}=\mathbb {R}[\varSigma ,{\mathrm {Sym}}^2(\varSigma ^*)]^W$ is a free A-module of rank $(n^2+n)/2$.

Third, $\varSigma$ is reducible as a W-representation if and only if $\varSigma =\varSigma _1\times \varSigma _2$ and $W=W_1\times W_2$ for two reflection groups $W_k\subset O(\varSigma _k)$—see section 2.2 in [14]. Thus the following proposition reduces the proofs of Theorem 4 and Observation 2 to the irreducible case.

Lemma 5

Let $W_k\subseteq O(\varSigma _k),$ $k=1,2$ be finite reflection groups in the Euclidean vector spaces $\varSigma _k,$ and let $W=W_1\times W_2\subset O(\varSigma )=O(\varSigma _1\times \varSigma _2)$. Then there are W-invariant polynomials on $\varSigma$ whose Hessians generate $\mathbb {R}[\varSigma ,{\mathrm {Sym}}^2(\varSigma ^*)]^W$ if and only if the same holds for $W_k\subseteq O(\varSigma _k),$ $k=1,2$.

Proof

Assume there are $Q_j\in \mathbb {R}[\varSigma ]^W$ whose Hessians generate $\mathbb {R}[\varSigma ,{\mathrm {Sym}}^2(\varSigma ^*)]^W$. Then the restrictions $Q_j|_{\varSigma _1}$ generate $\mathbb {R}[\varSigma _1,{\mathrm {Sym}}^2(\varSigma _1^*)]^{W_1}$ as an $\mathbb {R}[\varSigma _1]^{W_1}$-module.

Indeed, every $\sigma \in \mathbb {R}[\varSigma _1,{\mathrm {Sym}}^2(\varSigma _1^*)]^{W_1}$ can be naturally extended to ${\tilde{\sigma }}\in \mathbb {R}[\varSigma ,{\mathrm {Sym}}^2(\varSigma ^*)]^W$ that is constant on each copy of $\varSigma _2$. Then there are $a_j\in \mathbb {R}[\varSigma ]^W$ such that ${\tilde{\sigma }}=\sum _j a_j{\mathrm {Hess}}(Q_j)$. Therefore

$$\begin{aligned} \sigma =\sum _j (a_j|_{\varSigma _1}){\mathrm {Hess}}(Q_j|_{\varSigma _1}) \end{aligned}$$

and similarly for $\mathbb {R}[\varSigma _2,{\mathrm {Sym}}^2(\varSigma _2^*)]^{W_2}$.

For the converse, let $\rho _j\in \mathbb {R}[\varSigma _1]^{W_1}$, $j=1,\ldots , n_1$ and $\psi _j\in \mathbb {R}[\varSigma _2]^{W_2}$, $j=1,\ldots , n_2$ be basic invariants on $\varSigma _1$ and $\varSigma _2$ respectively; and $Q_j\in \mathbb {R}[\varSigma _1]^{W_1}$, $j=1,\ldots , (n_1^2+n_1)/2$, $R_j\in \mathbb {R}[\varSigma _2]^{W_2}$, $j=1,\ldots , (n_2^2+n_2)/2$ be homogeneous invariants whose Hessians form a basis for the corresponding spaces of equivariant symmetric 2-tensors.

We claim that the Hessians of the following set of $W=W_1\times W_2$-invariant polynomials on $\varSigma =\varSigma _1\times \varSigma _2$ form a basis for the space of equivariant symmetric 2-tensors on $\varSigma$:

$$\begin{aligned} \{Q_j\}\cup \{R_j\}\cup \{\rho _i\psi _j\} \end{aligned}$$

Indeed, $\mathbb {R}[\varSigma ,{\mathrm {Sym}}^2(\varSigma ^*)]^W$ decomposes as

$$\begin{aligned} \mathbb {R}[\varSigma ,{\mathrm {Sym}}^2(\varSigma _1^*)]^W\oplus \mathbb {R}[\varSigma ,{\mathrm {Sym}}^2(\varSigma _2^*)]^W\oplus \mathbb {R}[\varSigma ,\varSigma _1^*\otimes \varSigma _2^*]^W \end{aligned}$$

The first two pieces are freely generated over $\mathbb {R}[\varSigma ]^W$ by Hess$Q_j$ and Hess$R_j$. The third piece can be rewritten as $\mathbb {R}[\varSigma ,\varSigma _1^*\otimes \varSigma _2^*]^W=\mathbb {R}[\varSigma _1,\varSigma _1^*]^{W_1}\otimes \mathbb {R}[\varSigma _2,\varSigma _2^*]^{W_2}$. By Solomon’s Theorem [29], $\mathbb {R}[\varSigma _k,\varSigma _k^*]^{W_k}$, $k=1,2$, are freely generated by $d\rho _j$ and $d\psi _j$, so that $\mathbb {R}[\varSigma ,\varSigma _1^*\otimes \varSigma _2^*]^W$ is freely generated by $(d\rho _j\otimes d\psi _j + d\psi _j\otimes d\rho _j)$. To finish the proof of the claim one uses the product rule

$$\begin{aligned} {\mathrm {Hess}}(\rho _i\psi _j)= d\rho _i\otimes d\psi _j + d\psi _j\otimes d\rho _j+ \rho _i{\mathrm {Hess}}(\psi _j)+\psi _j{\mathrm {Hess}}(\rho _i) \end{aligned}$$

Proof of Theorem 4

By Lemma 5, we may assume W is irreducible of classical type.

If W is of type A, type B, or dihedral, then all multi-variable invariants are generated by polarizations, by [15, 34]. Hence it is enough to show that ${\mathcal {P}}^3\cap {\mathcal {M}}$ is generated, as an A-module, by Hessians of invariants. This follows from the product rule

$$\begin{aligned} {\mathrm {Hess}}(\rho _i\psi _j)= d\rho _i\otimes d\psi _j + d\psi _j\otimes d\rho _j+ \rho _i{\mathrm {Hess}}(\psi _j)+\psi _j{\mathrm {Hess}}(\rho _i) \end{aligned}$$

If, on the other hand, W is of type D, then the multi-variable invariants are generated by generalized polarizations—see Theorems 3.1 and 3.4 in [15], or Proposition 2 in Appendix 2 of [32]. But degree considerations imply that ${\mathcal {M}}$ is in fact generated by (classical) polarizations, hence also by Hessians by the product rule.

Calculation 1

Let W of type $H_3$, $H_4$, $F_4$, $E_6$, $E_7$ or $E_8$. Then there is a choice of basic invariants $\rho _1, \ldots \rho _n \text { with }\deg (\rho _1)<\cdots <\deg (\rho _n)$, and of a regular vector $v\in \varSigma$, such that $\{ {\mathrm {Hess}}(\rho ^*Q)(v)\ |\ Q\in T\}$ is linearly independent, where $\rho =(\rho _1, \ldots , \rho _n):\varSigma \rightarrow \mathbb {R}^n$, and T is the set of polynomials on $\mathbb {R}^n$ given in Table 1.

Table 1 Monomials with generating Hessians

We remark that the computation above is independent of the choice of basic invariants and regular vector—see Lemmas 6 and 7. Moreover, one may construct a set of basic invariants consisting of “orbit Chern classes” from some linear functional $\lambda _0:\varSigma \rightarrow \mathbb {R}$ and the degrees $d_i$. Namely, $\rho _i=\sum _{\lambda \in W\lambda _0}\lambda ^{d_i}$—see [9, 10, 22, 28]. See also [16, 21] for (other) explicit sets of basic invariants of exceptional groups. The degrees $d_1,\ldots ,d_n$ of the exceptional groups are listed in Table 2 for the convenience of the reader (see [4]).

Table 2 Degrees of exceptional finite reflection groups

The following Lemma is analogous to Proposition 3.13 in [14]. We will use the special case $U=\text {Sym}^2\varSigma ^*$ in proving both Observation 2 from Calculation 1, and independence of the choice of regular vector v in Calculation 1.

Lemma 6

Let $W\subset O(\varSigma )$ be an irreducible finite reflection group,  and $\eta : W\rightarrow O(U)$ an orthogonal representation,  with character $\chi$. Choose a basis $\{e_1,\ldots , e_l\}$ for U,  and let $f_1, \ldots , f_l\in \mathbb {R}[\varSigma , U]^W$ be homogeneous elements given by $f_i=\sum _j a_{ij} e_j$. Let $D=\det (a_{ij})\in \mathbb {R}[\varSigma ]$. Then : 

• (a) D is divisible by the following product over all reflections $r\in W{:}$

  $$\begin{aligned} J_\eta =\prod _{r}(\lambda _r)^\frac{l-\chi (r)}{2} \end{aligned}$$

  with $\lambda _r$ a linear functional whose kernel equals the hyperplane fixed by r.

• (b) If $\{f_i\}$ is a basis of $\mathbb {R}[\varSigma , U]^W$ over $\mathbb {R}[\varSigma ]^W,$ then D and $J_\eta$ have the same degree.

• (c) $\{f_i\}$ forms a basis if and only if $D=cJ_\eta$ for some $c\in \mathbb {R}-\{0\}$.

Proof

• (a) Let $r\in W$ be a reflection. The transformation $\eta (r):U\rightarrow U$ is diagonalizable with eigenvalues $1,-1$. In particular the multiplicity of $-1$ equals $k=(l-\chi (r))/2$. Assume, without loss of generality, that $e_1, \ldots , e_k$ is a basis for the eigenspace of $\eta (r)$ associated to the eigenvalue $-1$. Then the first k columns of $(a_{ij})$ are odd with respect to r. By multi-linearity, D vanishes to order k on the hyperplane fixed by r. In other words, D is divisible by $(\lambda _r)^k$. Since this is true for every reflection r, D is divisible by $J_\eta$.

• (b) For any graded vector space $E=\oplus _i E_i$, denote its Poincaré series by $P_t(E)=\sum _i \dim (E_i)t^i$. Let

  $$\begin{aligned} P(t)=\frac{P_t(\mathbb {R}[\varSigma ,{\mathrm {Sym}}^2(\varSigma ^*)]^{W})}{P_t(\mathbb {R}[\varSigma ]^W)} \end{aligned}$$

  Since $P(t)=\sum _{i=1}^lt^{\deg (f_i)}$, we have $\deg (D)=P'(1)$. On the other hand, $P_t(\mathbb {R}[\varSigma ]^W)=\varPi _{i=1}^n(1-t^{d_i})^{-1}$, while the numerator can be computed using Molien’s formula [18, page 249]:

  $$\begin{aligned} P_t(\mathbb {R}[\varSigma ,{\mathrm {Sym}}^2(\varSigma ^*)]^{W})=\frac{1}{|W|}\sum _{g\in W}\frac{\chi (g)}{\det (1-tg)} \end{aligned}$$

  Note that for $g=1$, $\det (1-tg)=(1-t)^n$; for $g=r$ a reflection, $\det (1-tg)=(1-t)^{n-1}(1+t)$; and for all other g, $(1-t)^{n-1}$ does not divide $\det (1-tg)$. Therefore, when computing $P'(1)$, terms of the latter type vanish:

  $$\begin{aligned}&|W|P'(1)=\chi (1)\left. \frac{d}{dt}\right| _{t=1} \frac{\varPi _{i=1}^n(1-t^{d_i})}{(1-t)^n} + \sum _{r \text { refl.}}\chi (r)\left. \frac{d}{dt}\right| _{t=1} \frac{\varPi _{i=1}^n(1-t^{d_i})}{(1-t)^{n-1}(1+t)}\\&P'(1)=\frac{1}{|W|}\left( \frac{lN|W|}{2} - \sum _{r \text { refl.}}\chi (r)\frac{|W|}{2}\right) =\sum _{r \text { reflection}}\frac{l-\chi (r)}{2} \end{aligned}$$

  where N is the number of reflections and we have used the identities $d_1\cdots d_n=|W|$ and $(d_1-1)+\cdots +(d_n-1)=N$. Thus $P'(1)$ equals the degree of $J_\eta$.

• (c) Assume $\{f_i\}$ is a basis. By parts 1 and 2, $D=cJ_\eta$ for some $c\in \mathbb {R}$. Assume for a contradiction that $c=0$. This means that $\{f_i(v)\}$ is linearly dependent for every $v\in \varSigma$. Take a regular v, and let B be a small open W-invariant neighborhood of the orbit Wv. Since W acts freely on B, on can construct $\sigma \in C^\infty (\varSigma , U)^W$ with supp$(\sigma )\subset B$, and $\sigma (v)\notin \text {span}\{f_i(v)\}$. This contradicts the fact that $\mathbb {R}[\varSigma , U]^W$ is dense in $C^\infty (\varSigma , U)^W$. Now assume $D=cJ_\eta$ for some $c\in \mathbb {R}-\{0\}$. Choose any homogeneous basis $\{f_i'\}$ of ${\mathcal {M}}$, and define $D'$ analogously to D. Writing $f_i=\sum _j b_{ij} f_j'$, we see that $\det (b_{ij})\in \mathbb {R}-\{0\}$, because $D=D'\det (b_{ij})$. This implies that $(b_{ij})$ is invertible in the algebra of matrices with coefficients in A, so that $\{f_i\}$ is a basis of ${\mathcal {M}}$ over A, too.

Proof of Observation 2

Assume Calculation 1. We claim that

$$\begin{aligned} \{f_i\}=\{\text {Hess}Q\ ,\ Q\in T\} \end{aligned}$$

forms a basis for ${\mathcal {M}}$ over A. Indeed, by inspection, the sum of the degrees of $f_i$ equals $\deg (J_\eta )$, which in this case is $N(n-1)$. Using Lemma 6, we see that $D=cJ_\eta$ for some $c\ne 0$, so that $\{f_i\}$ forms a basis.

Note that independence of the choice of regular vector follows from Lemma 6, because the zero set of $J_\eta$ is contained in the singular set.

Lemma 7

Calculation 1 is independent of the choice of basic invariants $\rho _i$.

Proof

Assume $\{\rho _i\}$ and $\{\psi _i\}$ are two sets of basic invariants, and that

$$\begin{aligned} s\{ {\mathrm {Hess}}(\rho ^*Q)(v)\ |\ Q\in T\} \end{aligned}$$

is linearly independent at some (hence all) regular vector $v\in \varSigma$. By Lemma 6, $\{ {\mathrm {Hess}}(\rho ^*Q)\ |\ Q\in T\}$ forms a basis of ${\mathcal {M}}$ as a free A-module. We claim that $\{ {\mathrm {Hess}}(\psi ^*Q)\ |\ Q\in T\}$ is also a basis, so that in particular $\{ {\mathrm {Hess}}(\psi ^*Q)(v)\ |\ Q\in T\}$ is linearly independent for every regular vector v.

Recall the graded version of Nakayama’s Lemma (see Exercise 4.6a in [6]): a set of homogeneous elements $f_i\in {\mathcal {M}}$ generates ${\mathcal {M}}$ as an A-module if and only if their images in ${\mathcal {M}}/I{\mathcal {M}}$ span it as real vector space, where I is the ideal of A generated by the elements of positive degree.

Since the degrees $d_i$ are all distinct, we may assume without loss of generality that $\psi _i=\rho _i + R_i(\rho _1, \ldots , \rho _{i-1})$. Note that the Hessian of a product of three or more (not necessarily distinct) basic invariants belongs to $I{\mathcal {M}}$, so that modulo $I{\mathcal {M}}$ we have $\text {Hess}(\psi _i \psi _j) \equiv \text {Hess}(\rho _i\rho _j)$, and

$$\begin{aligned} \text {Hess}(\psi _i)\equiv \text {Hess}(\rho _i)+ \sum _{j,k}c_{j,k}\text {Hess}(\rho _j \rho _k) \end{aligned}$$

where $c_{j,k}\in \mathbb {R}$ vanishes unless $d_j+d_k=d_i$.

By inspection, $y_jy_k\in T$ whenever $d_j+d_k=d_i$ for some i (see Table 2). Therefore $\{\text {Hess}(\psi ^*Q)\ ,\ Q\in T\}$ is written in terms of $\{\text {Hess}(\rho ^*Q)\ ,\ Q\in T\}$ (modulo $I{\mathcal {M}}$) using a triangular matrix with 1’s in the diagonal, showing that $\{{\mathrm {Hess}}(\psi ^*Q)\ |\ Q\in T\}$ is a basis of ${\mathcal {M}}$.