Equivalence of Moderate Growth Definitions on GL(2,R) and SL(2,R)

Definition 1.

Let $G$ be a matrix group. For $g\in G$, let $\Vert g{\Vert }_{HS}$ be the Hilbert-Schmidt norm defined by $\Vert g{\Vert }_{HS}^2=\mathrm{tr}(g^{t}g)$. A function $f:G\to \mathbb{C}$ is of moderate growth with respect to the Hilbert-Schmidt norm if there exist constants $C>0$ and $N\in \mathbb{N}$ such that for all $g\in G$,

$$|f(g)|\le C\Vert g{\Vert }_{HS}^N$$

Definition 2.

Let $G$ be a matrix group. For $g\in G$, let $\Vert g{\Vert }_{\text{max}}$ be the norm defined by $\Vert g{\Vert }_{\text{max}}=\max \{|g_{ij}|,|(g^{-1}{)}_{ij}|{\}}_{i,j}$. A function $f:G\to \mathbb{C}$ is of moderate growth with respect to the max-entry norm if there exist constants $C>0$ and $N\in \mathbb{N}$ such that for all $g\in G$,

$$|f(g)|\le C\Vert g{\Vert }_{\text{max}}^N$$

Theorem 1 (Equivalence Principle).

The two definitions of moderate growth are equivalent for a function $f:G\to \mathbb{C}$ if and only if the norms $\Vert \cdot {\Vert }_{HS}$ and $\Vert \cdot {\Vert }_{\text{max}}$ are polynomially equivalent on the matrix group $G$.

Proof. Suppose there exist constants $C_1,C_2>0$ and exponents $p_1,p_2\in \mathbb{N}$ such that for all $g\in G$, $\Vert g{\Vert }_{HS}\le C_1\Vert g{\Vert }_{\text{max}}^{p_1}$ and $\Vert g{\Vert }_{\text{max}}\le C_2\Vert g{\Vert }_{HS}^{p_2}$. If $f$ is of moderate growth with respect to $\Vert \cdot {\Vert }_{HS}$, then $|f(g)|\le C\Vert g{\Vert }_{HS}^N\le C(C_1\Vert g{\Vert }_{\text{max}}^{p_1}{)}^N=(C{C}_1^N)\Vert g{\Vert }_{\text{max}}^{N{p}_1}$. This shows $f$ is of moderate growth with respect to $\Vert \cdot {\Vert }_{\text{max}}$. The converse follows symmetrically by exchanging the roles of the norms. Thus, the equivalence of the definitions is equivalent to the polynomial equivalence of the norms. $\square$

Theorem 2 (Norm Equivalence on GL(2,R) and SL(2,R)).

(i) On $G=GL(2,\mathbb{R})$, the norms $\Vert \cdot {\Vert }_{HS}$ and $\Vert \cdot {\Vert }_{\text{max}}$ are not polynomially equivalent. (ii) On $G=SL(2,\mathbb{R})$, the norms $\Vert \cdot {\Vert }_{HS}$ and $\Vert \cdot {\Vert }_{\text{max}}$ are polynomially equivalent.

Proof. Let $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in GL(2,\mathbb{R})$. First, we establish an inequality in one direction. The definition of the max-entry norm implies $\max \{|a|,|b|,|c|,|d|\}\le \Vert g{\Vert }_{\text{max}}$.

$$\Vert g{\Vert }_{HS}^2=a^2+b^2+c^2+d^2\le 4(\max \{|a|,|b|,|c|,|d|\}{)}^2\le 4\Vert g{\Vert }_{\text{max}}^2$$

This yields $\Vert g{\Vert }_{HS}\le 2\Vert g{\Vert }_{\text{max}}$, which holds for any subgroup of $GL(2,\mathbb{R})$.

For the reverse inequality, we must bound $\Vert g{\Vert }_{\text{max}}$ by a polynomial in $\Vert g{\Vert }_{HS}$. This requires bounding the entries of both $g$ and $g^{-1}$. The entries of $g$ are bounded by its Hilbert-Schmidt norm, since $|g_{ij}{|}^2\le {\sum }_{k,l}|g_{kl}{|}^2=\Vert g{\Vert }_{HS}^2$, which implies $|g_{ij}|\le \Vert g{\Vert }_{HS}$. For the inverse,

$$g^{-1}=\frac{1}{\det g}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$$

The magnitudes of the entries of $g^{-1}$ are of the form $|g_{kl}|/|\det g|$. Since $|g_{kl}|\le \Vert g{\Vert }_{HS}$, any polynomial bound on $\Vert g{\Vert }_{\text{max}}$ relies on a polynomial upper bound for $1/|\det g|$ in terms of $\Vert g{\Vert }_{HS}$.

(i) For $G=GL(2,\mathbb{R})$, such a bound does not exist. Consider the sequence $g_n=\left(\begin{array}{cc}1/n& 0\\ 0& 1\end{array}\right)$ for $n\in \mathbb{N},n\ge 1$. The Hilbert-Schmidt norm is bounded: $\Vert g_n{\Vert }_{HS}^2=(1/n{)}^2+1^2\le 2$. However, $|\det g_n|=1/n$. Then $\Vert g_n{\Vert }_{\text{max}}=\max \{1/n,1,n,0\}=n$. The sequence $\Vert g_n{\Vert }_{\text{max}}$ is unbounded while $\Vert g_n{\Vert }_{HS}$ is bounded. No inequality of the form $\Vert g{\Vert }_{\text{max}}\le C_2\Vert g{\Vert }_{HS}^{p_2}$ can hold for all $g\in GL(2,\mathbb{R})$.

(ii) For $G=SL(2,\mathbb{R})$, we have $|\det g|=1$. The entries of $g^{-1}$ are the entries of the adjugate matrix, which are $\pm a,\pm b,\pm c,\pm d$. The maximum absolute entry of $g^{-1}$ is $\max \{|a|,|b|,|c|,|d|\}$. Therefore,

$$\Vert g{\Vert }_{\text{max}}=\max \{|g_{ij}|,|(g^{-1}{)}_{ij}|\}=\max \{|a|,|b|,|c|,|d|\}$$

Since $|g_{ij}|\le \Vert g{\Vert }_{HS}$ for all $i,j$, we have $\Vert g{\Vert }_{\text{max}}\le \Vert g{\Vert }_{HS}$. Combined with $\Vert g{\Vert }_{HS}\le 2\Vert g{\Vert }_{\text{max}}$, the norms are polynomially equivalent on $SL(2,\mathbb{R})$. $\square$