## Polynomials and Representations XXXII

We attempt to identify the irreducible rational representations of $G = GL_n\mathbb C.$ From the last article, we may tensor it with a suitable power of det and assume it is polynomial.

One key ingredient is the following rather ambiguous statement.

• Peter-Weyl Principle: any irrep can be embedded inside the ring of functions on G.

To make sense of this statement, we need to define a group action on the latter. This article is rather general; throughout, G is allowed to be any topological group and all representations are assumed to be continuous and finite-dimensional. ## C(G) as a Representation

Let C(G) be the ring of continuous functions $G\to \mathbb{C}.$

Definition. The action of $G\times G$ on $C(G)$ is defined as follows: $(x, y)\in G\times G, f \in C(G)\ \implies\ (x,y)\cdot f : G\to\mathbb{C}, g\mapsto f(x^{-1}gy).$

One checks that $(x',y')\cdot((x,y)\cdot f) = (x'x, y'y)\cdot f$.

Some examples of functions in C(G) are the matrix coefficients; recall that these are functions of the form $f : G \to \mathbb{C}, g\mapsto \alpha (\rho(v))$ for some representation $\rho : G\to GL(V)$, $v\in V$ and $\alpha \in V^\vee.$ A more economical way of writing this is $f = \beta\circ \rho$ for some $\beta \in \text{End}_{\mathbb C}(V)^\vee.$

Proposition. Let $\mathcal{O}(G)$ be the set of matrix coefficients. Then $\mathcal{O}(G)$ is a (unital) $\mathbb{C}$-subalgebra of $C(G)$.

Proof

To prove $1\in\mathcal{O}(G)$, let $\rho :G \to \mathbb{C}^*$ be the trivial representation.

Closure under addition and multiplication: suppose $\rho_i:G\to GL(V_i)$, $\beta_i\in \text{End}(V_i)^\vee$ for i = 1,2; let $f_i = \beta_i \circ \rho_i:G\to\mathbb{C}$ be the matrix coefficient. Taking: \begin{aligned} \rho_1 \oplus \rho_2 : G \to GL(V_1 \oplus V_2), \ \ &\beta_1 \oplus \beta_2 \in \text{End}(V_1)^\vee \oplus \text{End}(V_2)^\vee \subseteq \text{End}(V_1 \oplus V_2)^\vee, \\ \rho_1\otimes \rho_2 : G\to GL(V_1 \otimes V_2), \ \ &\beta_1 \otimes \beta_2 \in \text{End}(V_1)^\vee \otimes \text{End}(V_2)^\vee \cong \text{End}(V_1 \otimes V_2)^\vee,\end{aligned}

we have $f_1 + f_2 = (\beta_1 \oplus \beta_2) \circ (\rho_1 \oplus \rho_2)$ and $f_1 f_2 = (\beta_1 \otimes \beta_2) \circ(\rho_1 \otimes \rho_2).$ ♦ ## Identifying Matrix Coefficients in C(G)

Next we have:

Lemma. For any $f\in C(G)$, the following are equivalent.

1. The span of all $(g,g')\cdot f$ over all $(g, g')\in G \times G$ is finite-dimensional.
2. The span of all $(g,1)\cdot f$ over all $g\in G$ is finite-dimensional.
3. The span of all $(1,g)\cdot f$ over all $g\in G$ is finite-dimensional.
4. $f$ is a matrix coefficient.

Proof

That 1 ⇒ 2, 3 is obvious. Let us now prove 4 ⇒  1. Suppose $f = \beta\circ \rho$ for $\rho : G\to GL(V)$ and $\beta : \text{End}(V) \to \mathbb{C}$. Then the function $(x, y)f : G\to \mathbb{C}$ takes $g \mapsto f(x^{-1}gy) = \beta(\rho(x^{-1}) \rho(g) \rho(y)) = \beta'(\rho(g))$

where $\beta' : \text{End} (V) \to \mathbb{C}$ is a linear map which takes $\phi \mapsto \beta(\rho (x^{-1})\phi\rho(y))$. Hence the span of all $(x,y)f$ lies in $\{\beta' \circ \rho : \beta' \in \text{End}(V)^\vee\}$ and (1) follows since $\text{End}(V)^\vee$ is finite-dimensional.

Finally, we prove 3 ⇒ 4; the case of 2 ⇒ 4 is similar. Let V be the space spanned by all $(1, g)f$, so V is a finite-dimensional subspace of C(G) which is $(1\times G)$-invariant. Let

• $\rho : G \to GL(V)$ be given by the action of $(1\times G)$ on V;
• $\beta : \text{End}(V) \to \mathbb{C}$ be the map $\alpha \mapsto (\alpha(f))(e_G)$.

Now tracing through $\beta\circ \rho : G \to \mathbb{C}$ gives: $\overbrace{g}^{\in G}\ \mapsto \overbrace{(f_1\in V \mapsto (x\mapsto f_1(xg)))}^{\in GL(V)} \mapsto \left.(x\mapsto f(xg))\right|_{x=e} = f(g).$

Thus $f = \beta\circ \rho$ is a matrix coefficient. [Note: for 2 ⇒ 4, one needs to take the dual of V.]   ♦

Corollary. $\mathcal{O}(G)$ is the sum of all finite-dimensional $G\times G$-subrepresentations of $C(G)$. ## Weak Peter-Weyl Theorem

Let V be a representation of G; the vector space End(V) becomes a $G\times G$-rep via: $(x,y) : (\alpha : V\to V) \mapsto (\rho_V(x)\circ \alpha \circ\rho_V(y^{-1}) : V\to V).$

Another way of seeing this is: $\text{End}(V) \cong V\otimes V^\vee$ naturally and the action of $G\times G$ on the LHS is obtained from the RHS. Next, we obtain a linear map: $\text{End}(V)^\vee\to \mathcal{O}(G),\qquad \beta \mapsto (\beta \circ \rho_V : G\to \mathbb{C}).$ (*)

This is $G\times G$-equivariant:

• Indeed (xy) takes $\alpha\in \text{End}(V)$ to $\rho_V(x) \alpha \rho_V(y^{-1})$ so dually $\beta \in \text{End}(V)^\vee$ is taken to the function $(\alpha \in \text{End}(V)) \mapsto \beta(\rho_V(x^{-1})\alpha \rho_V(y)).$
• In $\mathcal{O}(G)$, (xy) takes $f:G\to\mathbb{C}$ to $g\mapsto f(x^{-1}gy).$ Letting $f = \beta\circ \rho_V$ it gets taken to $g \mapsto \beta(\rho_V(x^{-1}) \rho_V(g) \rho_V(y))$ so the following commutes: $\begin{matrix} \text{End}(V)^\vee &\longrightarrow & \mathcal{O}(G)\\ \downarrow & &\downarrow & \text{action of }(x,y)\\ \text{End}(V)^\vee &\longrightarrow &\mathcal{O}(G)\end{matrix}$ We have $\text{End}(V) \cong V \otimes V^\vee$ so its dual $V^\vee \otimes V$ is naturally isomorphic to End(V) as vector spaces and as G-reps, but not as $G\times G$-reps. This is because the isomorphism $V^\vee \otimes V \cong V\otimes V^\vee$ requires a twist of the two components, which is not $G\times G$-equivariant.

Now suppose G is compact Hausdorff.

Lemma. Let V, W be irreducible representations of G. Then:

• $\text{End}(V)$ is an irreducible representation of $G\times G$;
• $\text{End}(V) \cong \text{End}(W)$ as $G\times G$-reps if and only if $V\cong W$ as G-reps.

Proof

The character of End(V) is: $\chi_{\text{End}(V)}(g_1, g_2) = \chi_{V}(g_1) \chi_{V^\vee}(g_2).$ Taking the inner product: \begin{aligned}\int_{G\times G} \chi_{\text{End}(V)}(g_1, g_2) \chi_{\text{End}(W)}(g_1, g_2) d(g_1, g_2) &=\int_G \int_G \chi_{V}(g_1) \chi_{V^\vee}(g_2) \chi_{W}(g_1) \chi_{W^\vee}(g_2) dg_1 \, dg_2\\ &= \left(\int_G \chi_{V}(g_1) \chi_{W}(g_1) dg_1\right) \left(\int_G \chi_{V^\vee}(g_2) \chi_{W^\vee}(g_2) dg_2\right)\\&= \begin{cases} 1, \ \ &\text{if } V\cong W; \\ 0,\ \ &\text{otherwise.}\end{cases}\end{aligned}

This proves both claims together. ♦

Taking the direct sum of (*) over all irreps V of G, we obtain: $\displaystyle\bigoplus_{V \text{irrep of } G} \text{End}(V)^\vee\to \mathcal{O}(G).$

Weak Peter-Weyl Theorem. The above map is an isomorphism.

Proof

To show injectivity: the kernel of the map is a $G\times G$-submodule, so it is a direct sum of $G\times G$-irreps $\text{End}(V)^\vee.$ On the other hand, $\text{End}(V)^\vee \to \mathcal{O}(G)$ clearly cannot map to 0, since that would mean $\rho_V = 0.$

To show surjectivity: each $f\in \mathcal{O}(G)$ is, by definition, of the form $\beta \circ \rho$ for some $\rho : G \to GL(V)$, $\beta \in \text{End}(V)^\vee$. Thus lies in the image. ♦

We remind the reader that this theorem only works when G is compact Hausdorff.

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