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Thus, we can define the r-th cohomology group of $\varphi'$ modulo $K$ as the quotient

$$Z^r({\phi }^{\mathrm{\prime }},K)\mid d{C}^{r-1}({\phi }^{\mathrm{\prime }},K)\mathrm{.}Z^r(\backslash varphi\text{'},\; K)\; \backslash mid\; dC^\{r-1\}(\backslash varphi\text{'},\; K).$$

Summing up, we may say that $H^1(\varphi', K)$ measures the possible deformations $\lambda \to \varphi_\lambda$ of the homomorphism $\varphi$ such that the representation $\varphi_\lambda$ restricted to $K$ is fixed. There is obviously a homomorphism $H^r(\varphi', K) \to H^r(\varphi')$.

This is a standard construction in cohomology theory. Let $\varphi_K$ be the homomorphism $\varphi$ restricted to $K$. Let $\varphi_K$ be the corresponding homomorphism $K \to$ linear transformations on $L$.

Every cochain in $C^r(\varphi')$ defines by restriction to $K$ a cochain in $C^r(\varphi_K)$, hence also a linear map $H^r(\varphi') \to H^r(\varphi'_K)$. In certain dimensions, this is an exact sequence of the form:

$$\begin{array}{c}{H}^r({\phi }^{\mathrm{\prime }})\to H^r({\phi }_K^{\mathrm{\prime }})\to H^{r+1}({\phi }^{\mathrm{\prime }},K)\to H^{r+1}({\phi }^{\mathrm{\prime }})\to \dots \end{array}(3.1)\backslash begin\{array\}\{c\}\; H^r(\backslash varphi\text{'})\; \backslash to\; H^r(\backslash varphi\text{'}\_K)\; \backslash to\; H^\{r+1\}(\backslash varphi\text{'},\; K)\; \backslash to\; H^\{r+1\}(\backslash varphi\text{'})\; \backslash to\; \backslash dots\; \backslash end\{array\}\; \backslash quad\; (3.1)$$

("Exact sequence" means that the image of each of these homomorphisms is equal to the kernel of the succeeding one.) In practice, this is often used to compute $H^r(\varphi')$ in terms of $H^r(\varphi', K)$ and $H^r(\varphi'_K)$.

These constructions are of great interest for our program of computing $H^1(\varphi')$ for homomorphisms of $G$ arising from unitary representations since, as we will see, $H^1(\varphi', K)$ is more readily computable.

For example, suppose that

$$G=K\oplus P,\text{ with }[K,P]\subset P,[P,P]\subset K,G\; =\; K\; \backslash oplus\; P,\; \backslash text\{\; with\; \}\; [K,\; P]\; \backslash subset\; P,\; [P,\; P]\; \backslash subset\; K,$$

i.e., $K$ is a symmetric subalgebra of $G$.

Then, we have, for $\omega \in Z^1(\varphi', K)$, $X \in K$, $X(\omega) = 0 = X \rfloor d\omega + d(X \rfloor \omega) = 0$ hence

$$[\phi (X),\omega (Y)]=\omega ([X,Y])\text{for }X\in K,Y\in P\mathrm{.}(3.2)[\backslash varphi(X),\; \backslash omega(Y)]\; =\; \backslash omega([X,\; Y])\; \backslash quad\; \backslash text\{for\; \}\; X\; \backslash in\; K,\; Y\; \backslash in\; P.\; \backslash quad\; (3.2)$$

Thus, the set of operators $\omega(P)$ transforms under $\varphi(K)$ like the representation of $Ad_K$ in $P$. The cocycle condition is now

$$\omega ([X,Y])=[\phi (X),\omega (Y)]-[\phi (Y),\omega (X)]=0,\text{for }X,Y\in P\mathrm{.}(3.3)\backslash omega([X,\; Y])\; =\; [\backslash varphi(X),\; \backslash omega(Y)]\; -\; [\backslash varphi(Y),\; \backslash omega(X)]\; =\; 0,\; \backslash quad\; \backslash text\{for\; \}\; X,\; Y\; \backslash in\; P.\; \backslash quad\; (3.3)$$

As we shall see, at least for $G = SL(2, R)$, (3.2) and (3.3) seem to determine $H^1(\varphi, K)$ by purely algebraic means.

As we have seen, an $\omega \in Z^1(\varphi')$ induces a mapping of $U(G) \to U(L)$. Suppose that $L$ is a Lie algebra of skew-Hermitian operators on a Hilbert space $H$, and that $\varphi(G)$ is an irreducible family of operators on $H$. We constructed a homomorphism $H^1(\varphi') \to R^l$ by choosing a basis $\Delta_1, \dots, \Delta_l$ of Casimir operators for $G$, proving that for $\omega \in Z^1(\varphi')$, $\omega(\Delta_1), \dots, \omega(\Delta_l)$ are multiples $ib_1, \dots, ib_l$ of the identity operator in $H$, and mapping the cohomology class determined by $\omega$ into $(b_1, \dots, b_l) \in R^l$. We would like to get some idea of how to compute the image in $R^l$ of $H^1(\varphi', K)$.