2.1 Linear real reductive Lie group and Cartan subgroups

Let $G \subseteq GL(n, \mathbb {R})$ be a self-adjoint group which is also the group of real points of a connected algebraic group defined over $\mathbb {R}$ (we will additionally assume that G is connected in Section 4 and 5.). For brevity, we shall simply say that G is a linear real reductive Lie group. In this case, the Cartan involution on the Lie algebra $\mathfrak {g}$ is given by $\theta (X) = - X^T$ , where $X^{T}$ denotes the transpose matrix of X.

Let $K = G \cap O(n)$ , which is a maximal compact subgroup of G. Let $\mathfrak {k}$ be the Lie algebra of K. We have a $\theta $ -stable Cartan decomposition $\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {s}$ . Let H be a Cartan subgroup of G. Then, H has a $\theta $ -stable decomposition $H = T \times A$ , where $T = H \cap K$ is the compactly embedded part and $\exp \colon \mathfrak {a} \to A$ is a bijection. Here, the Lie algebra $\mathfrak {a}$ of A is an abelian subalgebra in $\mathfrak {s}$ . Any choice of a positive $\mathfrak {a}$ -root system defines a parabolic subalgebra $\mathfrak {p} = \mathfrak {m} + \mathfrak {a} + \mathfrak {n}$ in $\mathfrak {g}$ and thus defines a cuspidal parabolic subgroup $P = MAN$ in G. We say that P is the cuspidal parabolic subgroup associated to the Cartan subgroup H and vice versa.

Let $\mathfrak {h} = \mathfrak {t}\oplus \mathfrak {a}$ be an arbitrary $\theta $ -stable Cartan subalgebra of $\mathfrak {g}$ , where $\mathfrak {t} = \mathfrak {h} \cap \mathfrak {k}, \mathfrak {a} = \mathfrak {h} \cap \mathfrak {s}$ . Let $\mathcal {R}(\mathfrak {h}, \mathfrak {g})$ be the associated root system. If $\alpha \in \mathcal {R}(\mathfrak {h}, \mathfrak {g})$ is a real root (that is, $\alpha |_{\mathfrak {t}} \equiv 0$ ), then we can apply the Cayley transform to $\mathfrak {h}$ and obtain a new Cartan subalgebra $\mathfrak {h}' = \mathfrak {t}' \oplus \mathfrak {a}'$ such that $\dim (\mathfrak {t}')> \dim (\mathfrak {t})$ . Let P and $P'$ be the two cuspidal parabolic subgroups associated to $\mathfrak {h}$ and $\mathfrak {h}'$ , respectively. We say that $P'$ is more compact than P. In this way, we define a partial order on the set of all cuspidal parabolic subgroups of G.

2.2 Harish-Chandra’s Schwartz function

Following Harish-Chandra [Reference Harish-Chandra12, Lemma 2], we fix a real-valued nondegenerate bilinear symmetric form B on $\mathfrak {g}$ which is invariant under the adjoint action of G on $\mathfrak {g}$ and also the Cartan involution $\theta $ . Moreover, we can choose B such that $\langle , \rangle = -B(\cdot , \theta \cdot )$ defines a K-invariant inner product on $\mathfrak {s}$ . Thus, $\langle , \rangle $ induces a G-invariant Riemannian metric on $G/K$ . For $g \in G$ , we use $\|g\|$ to denote the Riemannian distance from $eK$ to $gK$ in $G/K$ . For every $n \geq 0, X, Y \in U(\mathfrak {g})$ , and $f \in C^{\infty }(G)$ , set

$$\begin{align*}\nu_{X, Y, n}(f) :=\sup _{g \in G}\Big\{(1+\|g\|))^{n} \Xi(g)^{-1} \big|L(X) R(Y) f(g)\big|\Big\}, \end{align*}$$

where L and R denote the left and right regular representations, respectively, and $\Xi $ is the Harish-Chandra’s $\Xi $ -function [Reference Harish-Chandra11].

The space $\mathcal {C}(G)$ is a Fréchet space in the semi- norms $\nu _{X, Y, n}$ . It is closed under convolution, which is a continuous operation on this space. Moreover, if G has equal rank (thus has discrete series representations), then all K-finite matrix coefficients of discrete series representations lie in $\mathcal {C}(G)$ . It is proved in [Reference Lafforgue19] that $\mathcal {C}(G)$ is a $*$ -subalgebra of the reduced group $C^{*}$ -algebra $C^{*}_r(G)$ that is closed under holomorphic functional calculus.

2.3 Cyclic cohomology

Definition 2.3. Let A be an algebra over $\mathbb {C}$ .Define the space of Hochschild cochains of degree k of A to be

$$\begin{align*}C^k(A) \colon = \mathrm{Hom}_{\mathbb{C}}\big(A^{\otimes(k+1)}, \mathbb{C}\big), \end{align*}$$

consisting of all bounded $k+1$ -linear functionals on A. Define the Hochschild codifferential $b \colon C^k(A) \to C^{k+1}(A)$ by the following formula:

$$\begin{align*}\begin{aligned} &b\Phi(a_0 \otimes \dots \otimes a_{k+1}) \\ =&\sum_{i=0}^k (-1)^i \Phi(a_0 \otimes \dots \otimes a_i a_{i+1} \otimes \dots \otimes a_{k+1}) + (-1)^{k+1} \Phi(a_{k+1}a_0 \otimes a_1 \otimes \dots \otimes a_{k}). \end{aligned} \end{align*}$$

The Hochschild cohomology of A is the cohomology of the complex $(C^{*}(A), b)$ .

Definition 2.4. We call a k-cochain $\Phi \in C^k(A)$ cyclic if for all $a_0, \dots , a_k \in A$ , it holds that

$$\begin{align*}\Phi(a_k, a_0, \dots, a_{k-1}) = (-1)^k \Phi(a_0, a_1, \dots, a_k). \end{align*}$$

The subspace $C^k_\lambda $ of cyclic cochains is closed under the Hochschild codifferential. The cyclic cohomology $HC^{*}(A)$ is defined by the cohomology of the subcomplex of cyclic cochains.

One can define a periodicity map $S: HC^{k}(A)\to HC^{k+2}(A)$ ; cf. [Reference Loday and Ronco20, Section 2.2.5]. And the periodic cyclic cohomology $HP^{*}(A)$ (for $*$ =even or odd) is defined to be the inductive limit of $HC^{2k}(A)$ (or $HC^{2k+1}(A)$ ). Let $R = (R_{i, j}), i, j = 1, \dots , m$ be an idempotent in $M_m(A)$ . The following formula

$$\begin{align*}\langle [\Phi], [R] \rangle \colon = \frac{1}{k!}\sum_{i_0,\cdots, i_{2k}=1}^m\Phi(R_{i_0i_1}, R_{i_1i_2}, ..., R_{i_{2k}i_0}) \end{align*}$$

defines a natural pairing between $[\Phi ]\in HP^{\text {even}}(A)$ and $K_0(A)$ ; that is,

$$\begin{align*}\langle \ \cdot \ , \ \cdot \ \rangle \colon HP ^{\text{even}}(A) \otimes K_0(A) \to \mathbb{C}. \end{align*}$$

3.1 Higher cyclic cocycles

Let $P = MAN$ be a cuspidal parabolic subgroup and denote $m = \dim A$ . By the Iwasawa decomposition, we have that

$$\begin{align*}G = KMAN. \end{align*}$$

We define a map $\widetilde {H} \colon G \to \mathfrak {a}$ by the decomposition

$$\begin{align*}g = \kappa (g)\mu(g) e^{\widetilde{H}(g)}n \in KM A N = G. \end{align*}$$

By fixing a basis for the Lie algebra $\mathfrak {a}$ , we write $\widetilde {H} = \left (H_1, \dots , H_m\right )$ .

Lemma 3.1. For any $g_0, g_1 \in G$ , the function $H_i\left (g_1 \kappa (g_0)\right )$ does not depend on the choice of $\kappa (g_0)$ . Moreover, the following identity holds:

$$\begin{align*}H_i(g_0) + H_i\left(g_1 \kappa(g_0)\right) = H_i(g_1 g_0). \end{align*}$$

Proof. Using $G = KMAN$ , we write

$$\begin{align*}g_0 = k_0m_0a_0 n_0, \hspace{5mm} g_1 = k_1m_1a_1 n_1. \end{align*}$$

Recall that the group $MA$ normalizes N and M commutes with A. Thus, for any $k \in K \cap M$ , there exists $n^{\prime }_1 \in N$ such that

$$\begin{align*}H_i(g_1 k) = H_i(k_1m_1a_1 n_1 k) = H_i(k_1 m_1ka_1 n_1' ) = H_i(a_1) = H_i(g_1). \end{align*}$$

It follows that $H_i \left (g_1 \kappa (g_0)\right )$ is well defined. Next, by the definition of $H_i$ ,

$$\begin{align*}H_i(g_1g_0) = H_i(a_1 n_1 k_0 a_0) = H_i(a_1 n_1 k_0) + H_i(a_0). \end{align*}$$

The lemma follows from the following identities:

$$\begin{align*}H_i(g_1 \kappa(g_0)) = H_i(a_1 n_1 k_0), \hspace{5mm} H_i(g_0) = H_i(a_0). \end{align*}$$

Let $S_m$ be the permutation group of m elements. For any $\tau \in S_m$ , let $\text {sgn}(\tau ) = \pm 1$ depending on the parity of $\tau $ .

Definition 3.2. We define a function

$$\begin{align*}C \in C^\infty\big(K \times G^{\times m}\big) \end{align*}$$

by

$$\begin{align*}\begin{aligned} C(k, g_1, \dots, g_m)\colon =& \sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot H_{\tau(1)}(g_1k) H_{\tau(2)}(g_2k ) \dots H_{\tau(m)}(g_m k)\\ =&\det \big(\widetilde{H}(g_1k), \dots, \widetilde{H}(g_mk) \big). \end{aligned} \end{align*}$$

Definition 3.3. For any $f_0, \dots , f_m \in \mathcal {C}(G)$ and semi-simple element $x \in M$ , we define a Hochschild cochain on $\mathcal {C}(G)$ by the following formula:

(3.1) $$ \begin{align} \begin{aligned} &\Phi_{P,x}(f_0, f_1, \dots, f_m)\colon =\int_{h \in M/Z_M(x)} \int_{K N} \int_{G^{\times m}} C(k, g_1g_2\dots g_m, \dots, g_{m-1}g_m, g_m) \\ & f_0 \big (kh x h^{-1}nk^{-1} (g_1\dots g_m)^{-1}\big)f_1(g_1) \dots f_m(g_m)dg_1\cdots dg_m dk dn dh, \end{aligned} \end{align} $$

where $Z_M(x)$ is the centralizer of x in M.

We prove in Theorem A.5 that the above integral (3.1) is convergent for all semi-simple elements $x \in M$ . A similar estimate leads us to the following property.

For simplicity, we omit the respective measures $dg_1, \cdots , dg_m$ , $dk$ , $dn$ , $dh$ , in the integral (3.1) for $\Phi _{P,x}$ .

Theorem 3.5. Let $P=MAN$ be a cuspidal parabolic subgroup of G and $x \in M$ be a semi-simple element. The cochain $\Phi _{P,x}$ is a cyclic cocycle and defines an element

$$\begin{align*}[\Phi_{P,x}] \in HC^m (\mathcal{C}(G)). \end{align*}$$

The proof of Theorem 3.5 occupies the rest of this section.

3.2 Cocycle condition

In this subsection, we prove that the cochain $\Phi _{P,x}$ introduced in Definition 3.3 is a Hochschild cocycle. We have the following expression for the codifferential of $\Phi _{P,x}$ :

(3.2) $$ \begin{align} \begin{aligned} &b\Phi_{P,x}(f_0, f_1, \dots, f_m, f_{m+1})\\ =&\sum_{i=0}^m (-1)^i \Phi_{P,x} \big(f_0, \dots, f_i \ast f_{i+1}, \dots, f_{m+1}\big) + (-1)^{m+1} \Phi_{P,x}\big(f_{m+1}\ast f_0, f_1, \dots, f_m \big). \end{aligned} \end{align} $$

Here, $f_i \ast f_{i+1}$ is the convolution product given by

$$\begin{align*}f_i \ast f_{i+1}(h) = \int_G f_i(g) f_{i+1}(g^{-1}h) dg. \end{align*}$$

Lemma 3.7. For $i = 0$ , the term on the right-hand side of (3.2) can be computed by the following integral:

$$\begin{align*}\begin{aligned} \Phi_{P,x} \big(f_0 \ast f_1, f_2 , \dots, f_{m+1}\big) &=\int_{M/Z_M(x)} \int_{K N} \int_{G^{\times (m+1)}} C(k, t_2t_3\dots t_{m+1}, \dots, t_mt_{m+1}, t_{m+1}) \\ &\quad f_0\big(kh x h^{-1}nk^{-1} (t_1\dots t_{m+1})^{-1}\big) f_1(t_1) f_2(t_2) \dots f_{m+1}(t_{m+1}). \end{aligned} \end{align*}$$

Proof. By definition,

$$\begin{align*}\begin{aligned} \Phi_{P,x} \big(f_0 \ast f_1, f_2 , \dots, f_{m+1}\big) &=\int_{M/Z_M(x)} \int_{K N} \int_G\int_{G^{\times m}} C(k, g_1g_2\dots g_m, \dots, g_{m-1}g_m, g_m) \\ &\quad f_0(g) f_1 \big (g^{-1} kh x h^{-1}nk^{-1} (g_1\dots g_m)^{-1}\big) f_2(g_1) \dots f_{m+1}(g_m). \end{aligned} \end{align*}$$

By changing variables,

$$\begin{align*}t_1 = g^{-1} kh x h^{-1}nk^{-1} (g_1\dots g_m)^{-1}, \hspace{5mm} t_j = g_{j-1}, \hspace{5mm} j = 2, \dots m+1, \end{align*}$$

we get

$$\begin{align*}g = kh x h^{-1}nk^{-1} (t_1\dots t_{m+1})^{-1}. \end{align*}$$

One can prove the lemma by replacing $g_i$ by $t_i$ .

Lemma 3.8. For $ 1\leq i \leq m$ , we have

(3.3)

where means that the term is omitted in the expression.

Proof. The left-hand side of the above equation,

(3.4) $$ \begin{align} \begin{aligned} &\Phi_{P,x} \big(f_0, \dots, f_i \ast f_{i+1}, \dots, f_{m+1}\big)\\ =&\int_{ M/Z_M(x)} \int_{K N} \int_G\int_{G^{\times m}} C(k, g_1g_2\dots g_m, \dots, g_{m-1}g_m, g_m) \\ &f_0 \big (khxh^{-1}nk^{-1} (g_1\dots g_m)^{-1}\big)f_1(g_1) \dots f_{i-1}(g_{i-1}) \\ &\big( f_i(g) f_{i+1} (g^{-1} g_i) \big) f_{i+2}(g_{i+1}) \dots f_{m+1}(g_m). \end{aligned} \end{align} $$

Let $t_j = g_j$ for $j = 1, \dots , i-1$ , and for $j = i+2, \dots , m+1$

$$\begin{align*}t_i = g, \hspace{5mm} t_{i+1} = g^{-1}g_i, \hspace{5mm} t_j = g_{j-1}. \end{align*}$$

Lemma 3.9. The last term in the right-hand side of (3.2) can be computed by the following integral:

$$\begin{align*}\begin{aligned} \Phi_{P,x} \big(f_{m+1} \ast f_0, f_1 , \dots, f_{m}\big)& =\int_{ M/Z_M(x)}\int_{K N} \int_{G^{\times (m+1)}} C(\kappa(t_{m+1}k), t_1t_2\dots t_m, \dots,t_{m-1}t_m, t_m) \\ &\quad f_0 \big (khxh^{-1}nk^{-1} (t_1\dots t_{m+1})^{-1}\big) f_1(t_1) \dots f_{m+1}(t_{m+1}). \end{aligned} \end{align*}$$

Proof. By definition,

(3.5) $$ \begin{align} \begin{aligned} \Phi_{P,x} \big(f_{m+1} \ast f_0, f_1 , \dots, f_{m}\big) &=\int_{ M/Z_M(x)}\int_{K N} \int_G\int_{G^{\times m}} C(k, g_1g_2\dots g_m, \dots, g_{m-1}g_m, g_m) \\ &\quad f_{m+1}(g) f_0 \big (g^{-1} k hxh^{-1}nk^{-1} (g_1\dots g_m)^{-1}\big) f_1(g_1) \dots f_{m}(g_m). \end{aligned} \end{align} $$

As before, we write

$$\begin{align*}t_j = g_{j}, \hspace{5mm} j = 1, \dots, m, \end{align*}$$

and $t_{m+1} = g$ . We can rewrite Equation (3.5) as

(3.6) $$ \begin{align} \begin{aligned} \Phi_{P,x} \big(f_{m+1} \ast f_0, f_1 , \dots, f_m\big) &=\int_{ M/Z_M(x)}\int_{K N} \int_{G^{\times (m+1)}} C(k, t_1t_2\dots t_m, \dots, t_{m-1}t_m, t_m) \\ &\quad f_0 \big (t_{m+1}^{-1} khxh^{-1}nk^{-1} (t_1\dots t_m)^{-1}\big) f_1(t_1) \dots f_{m+1}(t_{m+1}). \end{aligned} \end{align} $$

For all $t_{m+1} \in G$ and $k \in K$ , we decompose

$$\begin{align*}t_{m+1}^{-1} k = k_1 \mu_1 a_1 n_1 \in KMAN. \end{align*}$$

It follows that $k = t_{m+1} k_1 \mu _1 a_1n_1$ and $k = \kappa (t_{m+1} k_1)$ . We see

$$\begin{align*}\begin{aligned} &\Phi_{P,x} \big(f_{m+1} \ast f_0, f_1 , \dots, f_{m}\big) =\int_{ M/Z_M(x)}\int_{K N} \int_{G^{\times (m+1)}} C(k, t_1t_2\dots t_m, \dots, t_{m-1}t_m, t_m) \\ &f_0 \big ( k_1 \mu_1 a_1 n_1 hxh^{-1}nn_1^{-1}a_1^{-1}\mu_1^{-1} k_1^{-1} t_{m+1}^{-1} (t_1\dots t_m)^{-1}\big) f_1(t_1) \dots f_{m+1}(t_{m+1}). \end{aligned} \end{align*}$$

Since $\mu _1 a_1 \in MA$ normalizes the nilpotent group N, we can find $\tilde {n}_1, n^{\prime }_1 \in N$ such that

$$\begin{align*}\begin{aligned} &f_0 \big ( k_1 \mu_1 a_1 n_1 hxh^{-1}nn_1^{-1}a_1^{-1}\mu_1^{-1} k_1^{-1} t_{m+1}^{-1} (t_1\dots t_m)^{-1}\big)\\ &\quad =f_0 \big ( k_1 \mu_1 hx(\mu_1h)^{-1} \tilde{n}_1 n{n'}_1^{-1}k_1^{-1} (t_1\dots t_{m+1})^{-1}\big). \end{aligned} \end{align*}$$

Renaming $\tilde {n}_1 n{n'}_1^{-1}$ by n, we conclude that

$$\begin{align*}\begin{aligned} \Phi_{P,x} \big(f_{m+1} \ast f_0, f_1 , \dots, f_{m}\big)=&\int_{ M/Z_M(x)}\int_{K N} \int_{G^{\times (m+1)}} C(\kappa(t_{m+1} k_1), t_1t_2\dots t_m, \dots, t_{m-1}t_m, t_m) \\ &f_0 \big ( k_1 hxh^{-1}n k_1^{-1} (t_1\dots t_{m+1})^{-1}\big) \cdot f_1(t_1) \dots f_{m+1}(t_{m+1}).\\ \end{aligned} \end{align*}$$

This completes the proof.

Combining Lemmas 3.7, 3.8 and 3.9, we have reached the following equation:

(3.7) $$ \begin{align} \begin{aligned} & b\Phi_{P,x}(f_0, f_1, \dots, f_m, f_{m+1})\\ &\quad = \int_{ M/Z_M(x)} \int_{K N} \int_{G^{\times (m+1)}} \tilde{C}(k, t_1, \dots t_{m+1}) \cdot f_0 \big ( k_1 hxh^{-1}n k_1^{-1} (t_1\dots t_{m+1})^{-1}\big) \\ &\qquad \cdot f_1(t_1) \dots f_{m+1}(t_{m+1}), \end{aligned} \end{align} $$

where $\tilde {C} \in C^\infty \big (K \times G^{\times m}\big )$ is given by

Lemma 3.10. We have that

$$\begin{align*}b\Phi_{P,x}(f_0, f_1, \dots, f_m, f_{m+1}) = 0. \end{align*}$$

Proof. We will show that

$$\begin{align*}\tilde{C}(k, t_1, \dots, t_{m+1}) = 0. \end{align*}$$

To begin with, we notice the following expression:

$$\begin{align*}\begin{aligned} &C\big(\kappa(t_{m+1} k), t_1t_2\dots t_m, \dots, t_{m-1}t_m, t_m\big)\\ =&\sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot H_{\tau(1)}\big(t_1\dots t_m \kappa(t_{m+1} k)\big) H_{\tau(2)}\big(t_2\dots t_m \kappa(t_{m+1} k) \big) \dots H_{\tau(m)}\big(t_m \kappa(t_{m+1} k) \big). \end{aligned} \end{align*}$$

By Lemma 3.1, we have

$$\begin{align*}H_{\tau(i)}\big(t_i\dots t_m \kappa(t_{m+1} k) \big) = H_{\tau(i)}\big(t_i\dots t_{m+1} k \big) - H_{\tau(i)}(t_{m+1} k). \end{align*}$$

It follows that

$$\begin{align*}\begin{aligned} &C\big(\kappa(t_{m+1} k), t_1t_2\dots t_m, \dots, t_{m-1}t_m, t_m\big)\\ =&\sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot \Big(H_{\tau(1)}(t_1\dots t_{m+1} k) -H_{\tau(1)}(t_{m+1} k) \Big) \\ &\cdot \Big( H_{\tau(2)}(t_2\dots t_{m+1} k) - H_{\tau(2)}(t_{m+1} k) \Big)\dots \Big( H_{\tau(m)}(t_m t_{m+1} k) - H_{\tau(m)}(t_{m+1} k) \Big). \\ \end{aligned} \end{align*}$$

As the above sum is invariant with respect to the permutation group $S_m$ , the terms containing more than one factor $H_{\tau (i)}(t_{m+1}k)$ vanish. Thus,

$$\begin{align*}\begin{aligned} &C\big(\kappa(t_{m+1} k), t_1t_2\dots t_m, \dots, t_{m-1}t_m, t_m\big)\\ &\quad =\sum_{i=1}^m\sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot H_{\tau(1)}(t_1\dots t_{m+1} k) \dots \Big( - H_{\tau(i)}(t_{m+1} k) \Big) \dots H_{\tau(m)}(t_m t_{m+1} k) \\ &\qquad + \sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot H_{\tau(1)}(t_1\dots t_{m+1} k)\dots H_{\tau(m)}(t_m t_{m+1} k). \\ \end{aligned} \end{align*}$$

In the above expression, by changing the permutations

$$\begin{align*}\big(\tau(1), \dots, \tau(m)\big ) \mapsto \big(\tau(1), \dots , \tau(i-1), \tau(m), \tau(i), \dots ,\tau(m-1) \big), \end{align*}$$

we get

$$\begin{align*}\begin{aligned} &\sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot H_{\tau(1)}(t_1\dots t_{m+1} k) \dots H_{\tau(i-1)}(t_{i-1} \dots t_{m+1} k) \\ & \Big( - H_{\tau(i)}(t_{m+1} k) \Big) H_{\tau(i+1)}(t_{i+1} \dots t_{m+1} k) \dots H_{\tau(m)}(t_m t_{m+1} k)\\ =\, & (-1)^{n- i}\sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot H_{\tau(1)}(t_1\dots t_{m+1} k) \dots H_{\tau(i-1)}(t_{i-1}\dots t_m k) \\ & H_{\tau(m)}(t_{m+1} k) H_{\tau(i)}(t_{i+1} \dots t_{m+1} k) \dots H_{\tau(m-1)}(t_m t_{m+1} k). \end{aligned} \end{align*}$$

Putting all the above together, we have

and

We conclude from Lemmas 3.7–3.10 that $\Phi _{P,x}$ is a Hochschild cocycle. We will prove that $\Phi _{P,x}$ is cyclic in the next subsection.

3.3 Cyclic condition

In this subsection, we prove that the cocycle $\Phi _{P,x}$ is cyclic. Recall

(3.8) $$ \begin{align} \begin{aligned} \Phi_{P,x}(f_1, \dots, f_m, f_0) = & \int_{ M/Z_M(x)}\int_{K N} \int_{G^{\times m}} C(k, g_1g_2\dots g_m, \dots, g_{m-1}g_m, g_m) \\ &f_1 \big (khxh^{-1}nk^{-1} (g_1\dots g_m)^{-1}\big)f_2(g_1) \dots f_m(g_{m-1}) f_0(g_m). \end{aligned} \end{align} $$

By changing the variables,

$$\begin{align*}t_1 =khxh^{-1}nk^{-1} (g_1\dots g_m)^{-1}, \end{align*}$$

and $t_j = g_{j-1}$ for $j = 2, \dots , m$ . We have

$$\begin{align*}g_m = (t_1 \dots t_m)^{-1} khxh^{-1}nk^{-1}, \end{align*}$$

and

$$\begin{align*}g_i \dots g_m = (t_1 \dots t_i)^{-1} khxh^{-1}nk^{-1}. \end{align*}$$

It follows that

$$\begin{align*}\begin{aligned} &\Phi_{P,x}(f_1, \dots, f_m, f_0)= \int_{ M/Z_M(x)}\int_{K N} \int_{G^{\times m}}\\ & C\big(k, t_1^{-1} khxh^{-1}nk^{-1}, \dots, (t_1 \dots t_{m-1})^{-1} khxh^{-1}nk^{-1}, (t_1 \dots t_m)^{-1} khxh^{-1}nk^{-1} \big)\\ & f_0\big((t_1 \dots t_m)^{-1} khxh^{-1}nk^{-1}\big) f_1(t_1) f_2(t_2) \dots f_m(t_{m})\\ =& \sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot \int_{ M/Z_M(x)} \int_{K N} \int_{G^{\times m}} H_{\tau(1)}( t_1^{-1} khxh^{-1}n) \dots H_{\tau(m)}((t_1 \dots t_m)^{-1} khxh^{-1}n)\\ &f_0\big((t_1 \dots t_m)^{-1} khxh^{-1}nk^{-1}\big) f_1(t_1) f_2(t_2) \dots f_m(t_{m}).\\ \end{aligned} \end{align*}$$

We write

$$\begin{align*}(t_1 \dots t_m)^{-1} k = k_1 \mu_1 a_1 n_1 \in KMAN. \end{align*}$$

Then,

(3.9) $$ \begin{align} k = (t_1 \dots t_m) k_1 \mu_1 a_1 n_1, \end{align} $$

and

$$\begin{align*}\begin{aligned} f_0\big((t_1 \dots t_m)^{-1} khxh^{-1}nk^{-1}\big) &= f_0\big(k_1 \mu_1 a_1 n_1hxh^{-1} n n_1^{-1}a_1^{-1}\mu_1^{-1}k_1^{-1}(t_1 \dots t_m)^{-1} \big)\\ &= f_0\big(k_1h' x h^{\prime -1}n' k_1^{-1}(t_1 \dots t_m)^{-1} \big). \end{aligned} \end{align*}$$

Thus, we rewrite the right-hand side of (3.8)

(3.10) $$ \begin{align} \begin{aligned} \Phi_{P,x}(f_1, \dots, f_m, f_0)=& \sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot \int_{M/Z_M(x)} \int_{K N} \int_{G^{\times m}} H_{\tau(1)}( t_1^{-1} k) \dots H_{\tau(m)}((t_1 \dots t_m)^{-1} k)\\ &f_0\big( k_1hxh^{-1}nk_1^{-1}(t_1 \dots t_m)^{-1}\big) f_1(t_1) f_2(t_2) \dots f_m(t_{m}). \end{aligned} \end{align} $$

By Lemma 3.1 and (3.9), we have, for $1\leq i \leq m-1$ ,

(3.11) $$ \begin{align} \begin{aligned} H_{\tau(i)}\big( (t_1\dots t_i)^{-1} k\big)&=-H_{\tau(i)}\big( t_1\dots t_i \kappa((t_1\dots t_i)^{-1} k)\big)\\ &=-H_{\tau(i)}\big( t_1\dots t_i \kappa(t_{i+1} \dots t_m k_1)\big)\\ &= H_{\tau(i)}\big( t_{i+1} \dots t_m k_1 \big) - H_{\tau(i)}\big( t_1 \dots t_m k_1)\big), \end{aligned} \end{align} $$

and

(3.12) $$ \begin{align} \begin{aligned} H_{\tau(m)}\big( (t_1\dots t_m)^{-1} k\big)=&-H_{\tau(m)}\big( t_1\dots t_m \kappa((t_1\dots t_m)^{-1} k)\big)\\ =&-H_{\tau(m)}\big( t_1\dots t_m k_1\big). \end{aligned} \end{align} $$

Putting (3.10), (3.11) and (3.12) together, we see that

$$\begin{align*}\begin{aligned} &\Phi_{P,x}(f_1, \dots, f_m, f_0)\\ =& \sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot\int_{ M/Z_M(m)} \int_{K N} \int_{G^{\times m}} \prod_{i=1}^{m-1} \Big( H_{\tau(i)}\big( t_{i+1} \dots t_m k_1 \big) - H_{\tau(i)}\big( t_1 \dots t_m k_1)\big) \Big)\\ &\big(-H_{\tau(m)}( t_1\dots t_m k_1)\big)\cdot f_0\big( k_1hxh^{-1}nk_1^{-1}(t_1 \dots t_m)^{-1}\big) f_1(t_1) f_2(t_2) \dots f_m(t_{m}).\\ \end{aligned} \end{align*}$$

By symmetry, one can check that

(3.13) $$ \begin{align} \begin{aligned} &\sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot \prod_{i=1}^{m-1} \Big( H_{\tau(i)}\big( t_{i+1} \dots t_m k_1 \big) - H_{\tau(i)}\big( t_1 \dots t_m k_1)\big) \Big) \cdot H_{\tau(m)}( t_1\dots t_m k_1)\\ &\quad =\sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot \prod_{i=1}^{m-1} H_{\tau(i)}\big( t_{i+1} \dots t_m k_1 \big) \cdot H_{\tau(m)}( t_1\dots t_m k_1). \end{aligned} \end{align} $$

In the above expression, by changing the permutation

$$\begin{align*}\big(\tau(1), \dots, \tau(m)\big ) \mapsto \big(\tau(2), \dots ,\tau(m), \tau(1) \big) , \end{align*}$$

we can simplify Equation (3.13) to the following one:

$$\begin{align*}\begin{aligned} &\sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot \prod_{i=1}^{m-1} \Big( H_{\tau(i)}\big( t_{i+1} \dots t_m k_1 \big) - H_{\tau(i)}\big( t_1 \dots t_m k_1)\big) \Big) \cdot H_{\tau(m)}( t_1\dots t_m k_1)\\ &\quad =(-1)^{m-1} \cdot \sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot \prod_{i=1}^{m} H_{\tau(i)}\big( t_{i} \dots t_m k_1 \big). \end{aligned} \end{align*}$$

Finally, we have obtained the following identity:

$$\begin{align*}\begin{aligned} \Phi_{P,x}(f_1, \dots, f_m, f_0)&=(-1)^m\cdot \sum_{\tau \in S_m}\mathrm{sgn}(\tau)\cdot \int_{ M/Z_M(x)}\int_{K N} \int_{G^{\times m}} \prod_{i=1}^m H_{\tau(i)}\big( t_{i} \dots t_m k \big) \\ &\quad \cdot f_0\big( khxh^{-1}nk^{-1}(t_1 \dots t_m)^{-1}\big) f_1(t_1) f_2(t_2) \dots f_m(t_{m})\\ &=(-1)^m \cdot \Phi_{P,x}(f_0, \dots, f_m). \end{aligned} \end{align*}$$

Hence, we conclude that $\Phi _{P,x}$ is a cyclic cocycle, and we have completed the proof of Theorem 3.5.

4.1 Parabolic induction

A brief introduction to discrete series representations can be found in Appendix B. In this section, we review the construction of parabolic induction. Let H be a $\theta $ -stable Cartan subgroup of G with Lie algebra $\mathfrak {h}$ . Let $P= M_P A_P N_P$ be a cuspidal parabolic subgroup associated to H as in subsection 2.1.

Definition 4.1. Let $\eta $ be a unitary representation of $M_P$ and $\varphi $ a unitary representation of $A_P$ . The product $\sigma \otimes \varphi $ defines a unitary representation of $L_P=M_PA_P$ . A basic representation of G is a representation by extending $\sigma \otimes \varphi $ to P trivially across $N_P$ then inducing to G:

$$\begin{align*}\pi_{\eta, \varphi} = \operatorname{\mathrm{Ind}}_P^G(\eta \otimes \varphi). \end{align*}$$

If $\eta = \sigma $ is a discrete series representation, then $\operatorname {\mathrm {Ind}}_P^G(\sigma \otimes \varphi )$ will be called a basic representation induced from the discrete series representation of $M_P$ and unitary representation of $A_P$ . This construction is known as parabolic induction.

The character of $\pi _{\sigma , \varphi }$ is given in Theorem B.5, Equation (B.3) and Corollary B.6. Note that basic representations might not be irreducible. Knapp and Zuckerman completed the classification of tempered representations by showing which basic representations are irreducible and proved that every tempered representation of G is basic and every basic representation is tempered.

Now consider a single cuspidal parabolic subgroup $P \subseteq G$ with $L_P = M_PA_P$ , and form the group

$$\begin{align*}W(A_P, G) = N_K(\mathfrak{a}_P)/Z_K(\mathfrak{a}_P), \end{align*}$$

where $N_K(\mathfrak {a}_P)$ and $Z_K(\mathfrak {a}_P)$ are the normalizer and centralizer of $\mathfrak {a}_P$ in K, respectively. The group $W(A_P, G)$ acts as an outer automorphism of $M_P$ , and hence on the set of equivalence classes of representations of $M_P$ . For any discrete series representation $\sigma $ of $M_P$ , we define

$$\begin{align*}W_\sigma = \big\{ w \in N_K(\mathfrak{a}_P)\colon \operatorname{\mathrm{Ad}}_w^{*} \sigma \cong \sigma \big\} / Z_K(\mathfrak{a}_P). \end{align*}$$

Then, the above Weyl group acts on the family of induced representations

$$\begin{align*}\big\{\operatorname{\mathrm{Ind}}_P^G(\sigma \otimes \varphi)\big\}_{\varphi \in \widehat{A}_P}. \end{align*}$$

Definition 4.2. Let $P_1$ and $P_2$ be two cuspidal parabolic subgroups of G with $L_{P_i} = M_{P_i} A_{P_i}$ . Let $\sigma _1$ and $\sigma _2$ be two discrete series representations of $M_{P_i}$ . We say that

(4.1) $$ \begin{align} (P_1, \sigma_1) \sim (P_2, \sigma_2) \end{align} $$

if there exists an element w in G that conjugates $L_{P_1}$ of $P_1$ to $L_{P_2}$ of $P_2$ , and conjugates $\sigma _1$ to a representation unitarily equivalent to $\sigma _2$ . In this case, there is a unitary G-equivariant isomorphism

$$\begin{align*}\operatorname{\mathrm{Ind}}_{P_1}^G(\sigma_1 \otimes \varphi) \cong \operatorname{\mathrm{Ind}}_{P_2}^G(\sigma_2 \otimes (\operatorname{\mathrm{Ad}}_w^{*}\varphi)) \end{align*}$$

as Hilbert $C_0(\widehat {A}_{P_j})$ -modules that covers the isomorphism

$$\begin{align*}\operatorname{\mathrm{Ad}}_w^{*} \colon C_0(\widehat{A}_{P_1}) \to C_0(\widehat{A}_{P_2}). \end{align*}$$

We denote by $[P,\sigma ]$ the equivalence class of (4.1), and $\mathcal {P}(G)$ the set of all equivalence classes.

Finally, we recall the functoriality of parabolic induction.

Lemma 4.3. If $S = M_S A_S N_S$ is any cuspidal parabolic subgroup of L, then the unipotent radical of $SN_P$ is $N_SN_P$ , and the product

$$\begin{align*}Q = M_Q A_Q N_Q = M_S (A_sa:P) (N_SN_P) \end{align*}$$

is a cuspidal parabolic subgroup of G.

Proof. See [Reference Vogan29, Lemma 4.1.1].

Theorem 4.4 (Induction in stages)

Let $\eta $ be a unitary representation (not necessarily a discrete series representation) of $M_S$ . We decompose

$$\begin{align*}\varphi= (\varphi_1, \varphi_2) \in \widehat{A}_{S} \times \widehat{A}_P. \end{align*}$$

There is a canonical equivalence

$$\begin{align*}\operatorname{\mathrm{Ind}}_{P}^G\big(\operatorname{\mathrm{Ind}}^{M_P}_{S}(\eta \otimes \varphi_1) \otimes \varphi_2 \big)\cong \operatorname{\mathrm{Ind}}_{Q}^G\big(\eta \otimes (\varphi_1, \varphi_2)\big). \end{align*}$$

Proof. See [Reference Knapp18, p. 170].

4.2 Wave packets

Let $\widehat {G}_{\mathrm {temp}}$ be the set of equivalence classes of irreducible unitary tempered representations of G. For a Schwartz function f on G, its Fourier transform $\widehat {f}$ is defined by

$$\begin{align*}\widehat{f}(\pi) = \int_G f(g) \pi(g) dg, \hspace{5mm} \pi \in \widehat{G}_{\mathrm{temp}}. \end{align*}$$

Thus, the Fourier transform assigns to f a family of operators on different Hilbert spaces (tempered representations of G) indexed by $\pi $ .

The group $A_P$ , which consists entirely of positive definite matrices, is isomorphic to its Lie algebra via the exponential map. So $A_P$ carries the structure of a vector space, and we can speak of its space of Schwartz functions in the ordinary sense of harmonic analysis. The same goes for the unitary (Pontryagin) dual $\widehat {A}_P$ . By a tempered measure on $A_P$ , we mean a smooth measure for which integration extends to a continuous linear functional on the Schwartz space. Recall Harish-Chandra’s Plancherel formula for G [Reference Harish-Chandra13].

Theorem 4.5. There is a unique smooth, tempered, $W_\sigma $ -invariant function $m_{P, \sigma }$ on the spaces $\widehat {A}_P$ such that

$$\begin{align*}\|f\|_{L^{2}(G)}^{2}=\sum_{[P, \sigma] \in \mathcal{P}(G)} \int_{\widehat{A}_{P}}\left\|\widehat{f}(\pi_{\sigma, \varphi})\right\|_{HS}^{2} m_{P, \sigma}(\varphi) d\varphi \end{align*}$$

for every Schwartz function $f \in \mathcal {C}(G)$ . We call $m_{P, \sigma }(\varphi )$ the Plancherel density of the representation $\operatorname {\mathrm {Ind}}_P^G(\sigma \otimes \varphi )$ .

As $\varphi \in \widehat {A}_P$ varies, the induced G-representations

$$\begin{align*}\pi_{\sigma, \varphi} = \operatorname{\mathrm{Ind}}_P^G(\sigma \otimes \varphi) \end{align*}$$

can be identified with one another as representations of K. Denote by $\operatorname {\mathrm {Ind}}_P^G (\sigma )$ this common Hilbert space, and $\mathcal {L}(\operatorname {\mathrm {Ind}}_P^G( \sigma ))$ the space of K-finite Hilbert-Schmidt operators on $\operatorname {\mathrm {Ind}}_P^G( \sigma )$ . We shall discuss the adjoint to the Fourier transform.

Definition 4.6. Let h be a Schwartz-class function from $\widehat {A}_P$ into operators on $\operatorname {\mathrm {Ind}}_P^G (\sigma )$ that is invariant under the $W_\sigma $ -action. That is,

$$\begin{align*}h \in \big[\mathcal{C}(\widehat{A}_P) \otimes \mathcal{L}^2(\operatorname{\mathrm{Ind}}_P^G (\sigma)) \big]^{W_\sigma}. \end{align*}$$

The wave packet associated to h is the scalar function defined by the following formula:

$$\begin{align*}\check{h}(g) = \int_{\widehat{A}_P} \mathrm{Trace} \big(\pi_{\sigma, \varphi}(g^{-1}) \cdot h(\varphi) \big) \cdot m_{P, \sigma}(\varphi) d\varphi. \end{align*}$$

A fundamental theorem of Harish-Chandra asserts that wave packets are Schwartz functions on G.

4.3 Derivatives of Fourier transform

Let $P = MAN$ be a cuspidal parabolic subgroup. Here, P does not have to be maximal. Thus, we can decompose $A_P = A_\circ \times A_S$ (see Lemma 4.3). Suppose that $\pi = \operatorname {\mathrm {Ind}}_P^G(\eta ^{M} \otimes \varphi )$ , where $\eta ^{M}$ is an irreducible tempered representation (does not have to be a discrete series representation) of M with character denoted by $\Theta ^M(\eta ^M)$ and

$$\begin{align*}\varphi \in \widehat{A}_P = \widehat{A}_\circ \times \widehat{A}_S. \end{align*}$$

We denote $r = \dim (\widehat {A}_S)$ and $m = \dim (\widehat {A}_\circ )$ . As a vector space, let

$$\begin{align*}x_1, \dots, x_{\dim A_\circ}, x_{\dim A_\circ +1 }, \dots, x_{\dim A_P} \end{align*}$$

be the coordinates for $\widehat {A}_P$ . For $i = 0, \dots , m$ , let $h_i \in \mathcal {C}(\widehat {A}_P)$ , and $v_i, w_i$ be unit K-finite vectors in $\operatorname {\mathrm {Ind}}_P^G(\eta ^M)$ . We denote by $\frac {\partial h_i}{\partial j}$ the partial derivative of $h_i$ with respect to $x_j$ , $j = 1, \dots , m$ .

Definition 4.8. Suppose that $f_i\in \mathcal {C}(G)$ are wave packets associated to

$$\begin{align*}h_i \cdot v_{i} \otimes w_i^{*} \in \mathcal{C}\big(\widehat{A}_P, \mathcal{L}(\operatorname{\mathrm{Ind}}_P^G(\eta^M)\big). \end{align*}$$

We define an $(m+1)$ -linear map $T_\pi $ with an image in $\mathcal {C}(\widehat {A}_P)$ by

(4.2) $$ \begin{align} \begin{aligned} &T_\pi(\widehat{f}_0, \dots, \widehat{f}_m) \\ =& \begin{cases} \sum_{\tau \in S_m}\mathrm{sgn}(\tau) \cdot h_0(\varphi) \cdot \prod_{i=1}^m \frac{\partial h_i(\varphi)}{\partial_{\tau(i)}} & \text{if } \ v_i = w_{i+1}, i = 0, \dots, m-1, \ \text{and} \ v_m = w_0;\\ 0 & \text{otherwise}. \\ \end{cases} \end{aligned} \end{align} $$

Next we want to generalize the above definition to the Fourier transforms of all $f \in \mathcal {C}(G)$ . The induced space $\pi = \operatorname {\mathrm {Ind}}_P^G(\eta ^M \otimes \varphi )$ has a dense subspace:

(4.3) $$ \begin{align} \big\{s \colon K \to V^{\eta^M} \ \mathrm{continuous} \big| s(km) = \eta^M(m)^{-1}s(k) \ \mathrm{for} \ k \in K, m \in K \cap M \big\}, \end{align} $$

where $V^{\eta ^M}$ is the Hilbert space of M-representation $\eta ^M$ . The group G action on $\pi $ is given by the formula

(4.4) $$ \begin{align} \left( \pi\left(g\right) s \right)(k) = e^{-\langle \log \varphi + \rho, H(g^{-1}k)\rangle } \cdot \eta^M(\mu(g^{-1}k))^{-1} \cdot s(\kappa(g^{-1}k)), \end{align} $$

where $\rho $ denotes the half sum of positive roots. By Equation (4.4), the Fourier transform

$$\begin{align*}\begin{aligned} \left( \pi(f) s \right)(k) &= \left(\widehat{f}(\pi)s \right)(k) \\ &= \int_G(e^{-\langle \log \varphi + \rho, H(g^{-1}k)\rangle } \cdot \eta^M(\mu(g^{-1}k))^{-1} f(g) \cdot s(\kappa(g^{-1}k)) dg. \end{aligned} \end{align*}$$

Suppose now that $f_0, \dots , f_m $ are arbitrary Schwartz functions on G and $\widehat {f}_0,\dots , \widehat {f}_m$ are their Fourier transforms.

Definition 4.9. For any $1 \leq i \leq n$ , we define a bounded operator $\frac {\partial \widehat {f}}{\partial _i}$ from $\pi = \operatorname {\mathrm {Ind}}_P^G(\eta ^M \otimes \varphi )$ to itself by the following formula:

(4.5) $$ \begin{align} \left(\left(\frac{\partial\widehat{f}(\pi)}{\partial_i}\right)s \right)(k): = \int_G H_i(g^{-1}k) \cdot (e^{-\langle \log \varphi + \rho, H(g^{-1}k)\rangle } \cdot \eta^M(\mu(g^{-1}k))^{-1} \cdot f(g) \cdot s(\kappa(g^{-1}k)). \end{align} $$

We define an $(m+1)$ -linear map

$$\begin{align*}T_\pi\colon \underbrace{\mathcal{C}(G) \times \dots \times \mathcal{C}(G)}_{m+1} \to \mathbb{C} \end{align*}$$

by

$$\begin{align*}T_{\pi}(\widehat{f}_0, \dots, \widehat{f}_m) :=\sum_{\tau \in S_m}\mathrm{sgn}(\tau) \cdot \mathrm{Trace} \Big( \widehat{f}_0(\pi) \cdot \prod_{i=1}^m \frac{\partial \widehat{f}_i(\pi)}{\partial_{\tau(i)}} \Big). \end{align*}$$

The above definition generalizes (4.2).

Proposition 4.10. For any $\pi = \operatorname {\mathrm {Ind}}_P^G(\eta ^M \otimes \varphi )$ , we have the following identity:

(4.6) $$ \begin{align} \begin{aligned} T_{\pi}(\widehat{f}_0, \dots, \widehat{f}_m) &=(-1)^m\sum_{\tau \in S_m}\mathrm{sgn}(\tau)\int_{KMAN}\int_{G^{\times m} } H_{\tau(1)}\big(g_1 \dots g_m k\big)\dots H_{\tau(m)}\big(g_mk\big)\\ & e^{\langle \log \varphi + \rho, \log a\rangle } \cdot \Theta^M(\eta^M)(m) \cdot f_0(kmank^{-1} (g_1 g_2 \dots g_m)^{-1}) f_1(g_1) \dots f_m(g_m). \end{aligned} \end{align} $$

Proof. By definition, for any $\tau \in S_m$ ,

$$\begin{align*}\begin{aligned} &\Big( \widehat{f}_0(\pi) \cdot \prod_{i=1}^m \frac{\partial \widehat{f}_i(\pi)}{\partial_{\tau(i)}} \Big)s(k)\\ =& \int_{G^{\times {(k+1)}} } H_{\tau(1)}(g_1^{-1}\kappa(g_0^{-1}k)) H_{\tau(2)}\big(g_2^{-1}\kappa((g_0g_1)^{-1}k)\big)\\ & H_{\tau(m)}\big(g_m^{-1}\kappa((g_0g_1\dots g_{m-1})^{-1}k)\big)\cdot e^{-\langle \log \varphi+\rho, H((g_0g_1\dots g_m)^{-1}k) \rangle } \\ &\eta^M(\mu((g_0g_1\dots g_m)^{-1}k))^{-1} \cdot f_0(g_0) f_1(g_1) \dots f_m(g_m) s\big(\kappa((g_0g_1\dots g_m)^{-1}k)\big). \end{aligned} \end{align*}$$

By setting $g = (g_0g_1\dots g_m)^{-1}k$ , we have

$$\begin{align*}g_0= kg^{-1} (g_1 g_2 \dots g_m)^{-1}, \end{align*}$$

and

$$\begin{align*}(g_0 g_1 \dots g_j)^{-1} k = g_{j+1} g_{j+2} \dots g_m g. \end{align*}$$

Recall that recall $MA$ normalizes N and M centralizes A. We denote

$$\begin{align*}g^{-1} = \mu ank^{\prime -1} \in MA NK = G. \end{align*}$$

Thus,

(4.7) $$ \begin{align} \begin{aligned} &\left( \widehat{f}_0(\pi) \cdot \prod_{i=1}^m \frac{\partial \widehat{f}_i(\pi)}{\partial_{\tau(i)}} \right)s(k)\\ =&\int_{KMAN} \int_{G^{\times k} } H_{\tau(1)}\left(g_1^{-1}\kappa(g_1 \dots g_m k)\right) \dots H_{\tau(m)}\left(g_m^{-1}\kappa(g_mk)\right)\\ & e^{\langle \log \varphi+\rho, \log a\rangle} \cdot \eta^M(\mu) \cdot f_0\left(k\mu ank^{\prime -1} \left(g_1 g_2 \dots g_m\right)^{-1}\right) f_1(g_1) \dots f_m(g_m) s(k'). \end{aligned} \end{align} $$

By Lemma 3.1,

$$\begin{align*}H_{\tau(i)}\big(g_i^{-1}\kappa(g_i \dots g_m k)\big) = H_{\tau(i)}\big(g_{i+1} \dots g_m k\big) - H_{\tau(i)}\big(g_{i} \dots g_m k\big). \end{align*}$$

Thus,

(4.8) $$ \begin{align} \begin{aligned} &\sum_{\tau \in S_m}\mathrm{sgn}(\tau) H_{\tau(1)}\big(g_1^{-1}\kappa(g_1 \dots g_m k)\big) H_{\tau(2)}\big(g_2^{-1}\kappa((g_2g_3\dots g_mk)\big)\dots H_{\tau(m)}\big(g_m^{-1}\kappa(g_mk)\big)\\ =&\sum_{\tau \in S_m}\mathrm{sgn}(\tau) \big(H_{\tau(1)}(g_2 \dots g_m k) - H_{\tau(1)}(g_1g_2 \dots g_m k)\big)\big(H_{\tau(2)}(g_3 \dots g_m k) - H_{\tau(2)}(g_2 \dots g_m k)\big) \\ & \dots \big(H_{\tau(m-1)}(g_m k) - H_{\tau(m-1)}(g_{m-1}g_m k)\big) \big(- H_{\tau(m)}(g_mk) \big).\\ \end{aligned} \end{align} $$

By induction on m, we can prove that the right-hand side of Equation (4.8) equals

$$\begin{align*}(-1)^m \sum_{\tau \in S_m}\mathrm{sgn}(\tau) H_{\tau(1)}\big(g_1 \dots g_m k\big) H_{\tau(2)}\big(g_2g_3\dots g_mk\big)\cdot H_{\tau(m)}\big(g_mk\big). \end{align*}$$

By (4.7) and (4.8), we conclude that

$$\begin{align*}\begin{aligned} &\sum_{\tau \in S_m}\mathrm{sgn}(\tau) \Big( \widehat{f}_0(\pi) \cdot \prod_{i=1}^m \frac{\partial \widehat{f}_i(\pi)}{\partial_{\tau(i)}} \Big)s(k)\\ &\quad =(-1)^m \sum_{\tau \in S_m}\mathrm{sgn}(\tau) \int_{KMAN} \int_{G^{\times k} }H_{\tau(1)}\big(g_1 \dots g_m k\big) H_{\tau(2)}\big(g_2g_3\dots g_mk\big)\cdot H_{\tau(m)}\big(g_mk\big)\\ &\qquad e^{\langle \log \varphi+\rho, \log a \rangle} \cdot \eta^M(\mu) \cdot f_0\big(k\mu ank^{\prime -1} (g_1 g_2 \dots g_m)^{-1}\big) f_1(g_1) \dots f_m(g_m) s(k'). \end{aligned} \end{align*}$$

Expressing it as a kernel operator, we have

$$\begin{align*}\sum_{\tau \in S_m}\mathrm{sgn}(\tau) \cdot \Big( \widehat{f}_0(\pi) \cdot \prod_{i=1}^m \frac{\partial \widehat{f}_i(\pi)}{\partial_{\tau(i)}} \Big)s(k) = \int_{K} L(k, k') s(k') dk', \end{align*}$$

where

$$\begin{align*}\begin{aligned} L(k, k') &= (-1)^m \sum_{\tau \in S_m}\mathrm{sgn}(\tau)\int_{MAN}\int_{G^{\times k} } H_{\tau(1)}\big(g_1 \dots g_m k\big) H_{\tau(2)}\big(g_2g_3\dots g_mk\big)\cdot H_{\tau(m)}\big(g_mk\big)\\ &\quad e^{\langle \log \varphi+\rho, \log a \rangle} \cdot \eta^M(\mu) \cdot f_0(k\mu ank^{\prime -1} (g_1 g_2 \dots g_m)^{-1}) f_1(g_1) \dots f_m(g_m). \end{aligned} \end{align*}$$

The proposition follows from the fact that $T_{\pi } = \int _K L(k, k) dk$ .

Suppose that $P_1= M_1 A_1 N_1$ and $P_2 = M_2 A_2 N_2$ are two cuspidal parabolic subgroups such that $P_1$ is more noncompact than $P_2$ . Moreover, we assume that the induced representation $\operatorname {\mathrm {Ind}}_{P_1}^G\big (\sigma _1 \otimes \varphi _1\big )$ is reducible and decomposes into

$$\begin{align*}\operatorname{\mathrm{Ind}}_{P_1}^G\big(\sigma_1 \otimes \varphi_1\big) = \bigoplus_k \operatorname{\mathrm{Ind}}_{P_2}^G\big(\delta_k \otimes \varphi_2\big), \end{align*}$$

where $\sigma _1$ is a discrete series representation of $M_1$ and $\delta _k$ are different limit of discrete series representations of $M_2$ . We decompose

$$\begin{align*}\widehat{A}_2 = \widehat{A}_\circ \times \widehat{A}_S, \quad \widehat{A}_1 = \widehat{A}_2 \times \widehat{A}_{12} = \widehat{A}_\circ \times \widehat{A}_S \times \widehat{A}_{12}. \end{align*}$$

Let $h_i \in \mathcal {C}(\widehat {A}_1 )$ , and $v_i, w_i$ be unit K-finite vectors in $\operatorname {\mathrm {Ind}}_{P_1}^G(\sigma _1)$ for $i = 0, \dots , m$ . We put

$$\begin{align*}\widehat{f}_i = h_i \cdot v_{i} \otimes w_i^{*} \in \mathcal{C}\big(\widehat{A}_1, \mathcal{L}(\operatorname{\mathrm{Ind}}_{P_1}^G(\sigma_1)\big). \end{align*}$$

The following lemma follows from Definition 4.9.

Lemma 4.11. Suppose that $\pi = \operatorname {\mathrm {Ind}}_{P_2}^G\big (\delta _k \otimes \varphi _2\big )$ . If

$$\begin{align*}v_i = w_{i+1}, \quad i= 0, \dots, m-1, \end{align*}$$

and $v_m = w_0$ , then

$$\begin{align*}T_\pi(\widehat{f}_0 , \dots, \widehat{f}_m) = \sum_{\tau \in S_m}\mathrm{sgn}(\tau) \cdot h_0\big((\varphi_2, 0)\big) \cdot \prod_{i=1}^m \frac{\partial h_i\big((\varphi_2, 0)\big)}{\partial_{\tau(i)}}. \end{align*}$$

Otherwise, $T_\pi (\widehat {f}_0 , \dots , \widehat {f}_m) = 0$ .

4.4 Cocycles on $\widehat {G}_{\mathrm {temp}}$

Let $P_\circ = M_\circ A_\circ N_\circ $ be a maximal cuspidal parabolic subgroup (cf. Definition 2.1) and T be the maximal torus of K. In particular, the Lie algebra $\mathfrak {t}$ of T gives a Cartan subalgebra of $\mathfrak {m}_\circ $ and $T \subseteq M_\circ $ .

Definition 4.12. For an irreducible tempered representation $\pi $ of G, we define

$$\begin{align*}\mathcal{A}(\pi) = \left\{\eta^{M_\circ}\otimes \varphi \in (\widehat{M_\circ A_\circ})_{\mathrm{temp}} \Big| \operatorname{\mathrm{Ind}}^G_{P_\circ}(\eta^{M_\circ}\otimes \varphi) = \pi\right\}. \end{align*}$$

Definition 4.13. Let $m(\eta ^{M_\circ })$ be the Plancherel density for the irreducible tempered representations $\eta ^{M_\circ }$ of $M_\circ $ . We put

$$\begin{align*}\mu\big(\pi \big) = \sum_{\eta^{M_\circ}\otimes \varphi \in \mathcal{A}(\pi)} m(\eta^{M_\circ}). \end{align*}$$

Recall the Plancherel formula

(4.9) $$ \begin{align} f(e) =\int_{\pi \in \widehat{G}_{\mathrm{temp}}} \mathrm{Trace}\big(\widehat{f}(\pi) \big)\cdot m(\pi)d\pi, \end{align} $$

where $m(\pi )$ is the Plancherel density for the G-representation $\pi $ .

Definition 4.14. We define $\widehat {\Phi }_{e}$ by the following formula:

$$\begin{align*}\widehat{\Phi}_{e}(\widehat{f}_0 , \dots, \widehat{f}_m) = \int_{\pi \in \widehat{G}_{\mathrm{temp}}} T_{\pi}(\widehat{f}_0, \dots, \widehat{f}_m) \cdot \mu(\pi) \cdot d\pi. \end{align*}$$

Theorem 4.15. For any $f_0, \dots , f_m \in \mathcal {C}(G)$ ,

$$\begin{align*}\Phi_{P_\circ,e}(f_0, \dots, f_m) =(-1)^m\widehat{\Phi}_{e}(\widehat{f}_0, \dots, \widehat{f}_m). \end{align*}$$

The proof of Theorem 4.15 is presented in Section 4.5.

Example 4.16. Suppose that $G = \mathbb {R}^m$ . Let

$$\begin{align*}x^i = (x^i_1, \dots x^i_{m}) \in \mathbb{R}^m \end{align*}$$

be the coordinates of $\mathbb {R}^{m}$ . On $\mathcal {C}(\mathbb {R}^m)$ , the cocycle $\Phi _{P_\circ ,e}$ is given as follows:

$$\begin{align*}\begin{aligned} &\Phi_{P_\circ,e}(f_0, \dots, f_{m})\\ =& \sum_{\tau \in S_{m}} \mathrm{sgn}(\tau) \int_{x^1 \in \mathbb{R}^{m}} \dots \int_{x^{m}\in \mathbb{R}^{m}} x^1_{\tau(1)} \dots x^{m}_{\tau(m)}f_0\big(-(x^1 +\dots+ x^{m}) \big)f_1(x^1) \dots f_{m}(x^{m}). \end{aligned} \end{align*}$$

However, the cocycle $\widehat {\Phi }_e$ on $\mathcal {C}(\widehat {\mathbb {R}}^m)$ is given as follows:

$$\begin{align*}\widehat{\Phi}_e(\widehat{f}_0, \dots, \widehat{f}_{m}) =(-\sqrt{-1})^m \int_{\mathbb{R}^{m}} \widehat{f}_0 d \widehat{f}_1 \dots d\widehat{f}_{m}. \end{align*}$$

To see they are equal, we can compute

To introduce the cocycle $\widehat {\Phi }_t$ for any $t \in T^{\text {reg}}$ , we first recall the formula (C.7) for orbital integrals splits into three parts:

$$\begin{align*}\text{regular part} + \text{singular part} + \text{higher part}. \end{align*}$$

Accordingly, for any $t \in T^{\text {reg}}$ , we define

• • regular part: for regular $\lambda \in \Lambda ^{*}_{K} + \rho _c$ (see Definition B.1), we define

  $$\begin{align*}\begin{aligned} &\left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\lambda} = \left(\sum_{w \in W_{K}} (-1)^w \cdot e^{w \cdot \lambda}(t) \right) \cdot \int_{\varphi \in \widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(\lambda)\otimes \varphi)}(\widehat{f}_0 , \dots, \widehat{f}_m)\cdot d\varphi, \end{aligned} \end{align*}$$
  where $\sigma ^{M_\circ }(\lambda )$ is the discrete series representation of $M_\circ $ with Harish-Chandra parameter $\lambda $ .

• • singular part: for any singular $\lambda \in \Lambda ^{*}_{K} + \rho _c$ , we define

  $$\begin{align*}\begin{aligned} &\left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\lambda} = \frac{\sum_{w \in W_{K}} (-1)^w \cdot e^{w \cdot \lambda}(t) }{n(\lambda)} \cdot \sum_{i=1}^{n(\lambda)} \int_{\varphi \in \widehat{A}_\circ} \epsilon(i) \cdot T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}_i(\lambda)\otimes \varphi)}(\widehat{f}_0 , \dots, \widehat{f}_m)\cdot d\varphi,\\ \end{aligned} \end{align*}$$
  where $\sigma ^{M_\circ }_i$ are limit of discrete series representations of $M_\circ $ with Harish-Chandra parameter $\lambda $ and $n(\lambda )$ is the number of different limit of discrete series representations with Harish-Chandra parameter $\lambda $ , and $\epsilon (i) = 1$ for $i = 1, \dots \frac {n(\lambda )}{2}$ and $\epsilon (i) = -1$ for $i = \frac {n(\lambda )}{2}+1, \dots n(\lambda )$ (compare with the notations in Theorem C.7).

• • higher part:

  $$\begin{align*}\begin{aligned} &\left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\mathrm{high}}=\int_{\pi \in \widehat{G}^{\text{high}}_{\mathrm{temp}}} T_{\pi}(\widehat{f}_0, \dots, \widehat{f}_m)\cdot \left( \sum_{\eta^{M_\circ} \otimes \varphi \in \mathcal{A}(\pi)} \kappa^{M_\circ} (\eta^{M_\circ}, t) \right) \cdot d\varphi, \end{aligned} \end{align*}$$
  where the functions $\kappa ^{M_\circ }(\eta ^{M_\circ }, t)$ are defined in Subsection C.3, and
  $$\begin{align*}\widehat{G}^{\text{high}}_{\mathrm{temp}} = \big\{\pi \in \widehat{G}_{\mathrm{temp}} \big| \pi =\operatorname{\mathrm{Ind}}_{P_\circ}^G( \eta^{M_\circ} \otimes \varphi), \eta^{M_\circ} \in (\widehat{M}_\circ)^{\text{high}}_{\mathrm{temp}} \big\}, \end{align*}$$
  where $(\widehat {M}_\circ )^{\text {high}}_{\mathrm {temp}}$ is the set of irreducible tempered representations of $M_\circ $ which are not (limit of) discrete series representations.

Definition 4.17. For any element $t \in T^{\text {reg}}$ , we define

$$\begin{align*}\begin{aligned} &\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m) \\ =& \sum_{\text{regular} \ \lambda \in \Lambda^{*}_{K}+ \rho_c} \left[\widehat{\Phi}_{h}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\lambda} + \sum_{\text{singular} \ \lambda \in \Lambda^{*}_{K}+ \rho_c} \left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\lambda}+ \left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\mathrm{high}}. \end{aligned} \end{align*}$$

Theorem 4.18. For any $t \in T^{\text {reg}}$ , and $f_0, \dots , f_m \in \mathcal {C}(G)$ ,

$$\begin{align*}\Delta_{T}^{M_\circ}(t) \Phi_{P_\circ,t}(f_0, \dots, f_m) = (-1)^m \widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m). \end{align*}$$

The proof of Theorem 4.18 is presented in Section 4.6.

4.5 Proof of Theorem 4.15

We split the proof into several steps:

$\mathbf {Step \ 1} \colon $ Change the integral from $\widehat {G}_{\text {temp}}$ to $(\widehat {M_\circ A_\circ } )_{\text {temp}}$ :

$$\begin{align*}\begin{aligned} \widehat{\Phi}_{e}(\widehat{f}_0, \dots, \widehat{f}_m) &= \int_{\pi \in \widehat{G}_{\mathrm{temp}}} T_{\pi}(\widehat{f}_0, \dots, \widehat{f}_m) \cdot \mu(\pi) \cdot d\pi \\ &= \int_{\eta^{M_\circ} \otimes \varphi \in (\widehat{M_\circ A_\circ} )_{\text{temp}}} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\eta^{M_\circ} \otimes \varphi)}(\widehat{f}_0 , \dots, \widehat{f}_m)\cdot m(\eta^{M_\circ}).\\ \end{aligned} \end{align*}$$

$\mathbf {Step \ 2} \colon $ Replace $T_{\operatorname {\mathrm {Ind}}_{P_\circ }^G(\eta ^{M_\circ } \otimes \varphi )}$ in the above expression of $\widehat {\Phi }_e$ by Equation (4.6):

$$\begin{align*}\begin{aligned} &\widehat{\Phi}_{e}(\widehat{f}_0 , \dots, \widehat{f}_m) \\ &\quad =(-1)^m\sum_{\tau \in S_m}\mathrm{sgn}(\tau) \int_{\eta^{M_\circ}\otimes \varphi \in (\widehat{M_\circ A_\circ} )_{\text{temp}}} \int_{KM_\circ A_\circ N_\circ}\int_{G^{\times m} } H_{\tau(1)}\big(g_1 \dots g_m k\big)\dots H_{\tau(m)}\big(g_mk\big)\\ &\qquad e^{\langle \log \varphi+\rho, \log a\rangle} \cdot \Theta^{M_\circ}(\eta^{M_\circ})(m) \cdot f_0\left(kmank^{-1} (g_1 g_2 \dots g_m)^{-1}\right) f_1(g_1) \dots f_m(g_m) \cdot m(\eta^{M_\circ}). \\ \end{aligned} \end{align*}$$

$\mathbf {Step \ 3} \colon $ Simplify $\widehat {\Phi }_e$ by Harish-Chandra’s Plancherel formula. We write

$$\begin{align*}\begin{aligned} & \widehat{\Phi}_{e}(\widehat{f}_0 , \dots, \widehat{f}_m) \\ &\quad =(-1)^m\sum_{\tau \in S_m}\mathrm{sgn}(\tau) \int_{K N_\circ}\int_{G^{\times m} } H_{\tau(1)}\left(g_1 \dots g_m k\right)\dots H_{\tau(m)}\left(g_mk\right) \cdot f' \cdot f_1(g_1) \dots f_m(g_m),\\ \end{aligned} \end{align*}$$

where the function $f'$ is defined by the following formula:

$$\begin{align*}\begin{aligned} f'(k, n, g_1, \dots, g_m)= &\int_{\eta^{M_\circ} \otimes \varphi \in (\widehat{M_\circ A_\circ} )_{\text{temp}}} \int_{M_\circ A_\circ } e^{\langle \log \varphi+\rho, \log a\rangle} \\ &\Theta^{M_\circ}(\eta^{M_\circ})(m) \cdot f_0(kmank^{-1} (g_1 g_2 \dots g_m)^{-1}) \cdot m(\eta^{M_\circ}). \end{aligned} \end{align*}$$

If we put

$$\begin{align*}c(k, m, a, n, g_1, \dots, g_m)= e^{\langle \rho, \log a \rangle}\cdot f_0\left(kmank^{-1} (g_1 g_2 \dots g_m)^{-1}\right), \end{align*}$$

then

$$\begin{align*}f'(k, n, g_1, \dots, g_m) = \int_{\eta^{M_\circ}\otimes \varphi \in (\widehat{M_\circ A_\circ} )_{\text{temp}}} \left( \Theta^{M_\circ}(\eta^{M_\circ}) \otimes \varphi \right)(c) \cdot m(\eta^{M_\circ}). \end{align*}$$

By (4.9),

$$\begin{align*}f' = c(k, e, e, n, g_1, \dots, g_m) = f_0\left(knk^{-1} (g_1 g_2 \dots g_m)^{-1}\right). \end{align*}$$

This completes the proof.

4.6 Proof of Theorem 4.18

Our proof strategy for Theorem 4.18 is similar to the one used to prove Theorem 4.15. We split its proof into 3 steps as before.

$\mathbf {Step \ 1} \colon $ Let $\Lambda ^{*}_T$ be the weight lattice for T and $\Lambda ^{*}_{K \cap M_\circ }$ be the intersection of $\Lambda ^{*}_T$ and the positive Weyl chamber for the group $M_\circ \cap K$ . We denote by $\rho _c^{M_\circ \cap K}$ the half sum of positive roots for $M_\circ \cap K$ . For any $\lambda \in \Lambda ^{*}_{K \cap M_\circ } $ , we can find an element $w \in W_K /W_{K \cap M_\circ }$ such that $w \cdot \lambda \in \Lambda ^{*}_{K}$ . Moreover, for any $w \in W_K /W_{K \cap M_\circ }$ ,

(4.10) $$ \begin{align} \operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(\lambda)\otimes \varphi) \cong \operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(w \cdot \lambda)\otimes \varphi). \end{align} $$

For the regular part,

(4.11) $$ \begin{align} \begin{aligned} & \sum_{\text{regular} \ \lambda \in \Lambda^{*}_{K} + \rho_c} \left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\lambda}\\ =& \sum_{\text{regular} \ \lambda \in \Lambda^{*}_{K}+ \rho_c} \left(\sum_{w \in W_{K}} (-1)^w \cdot e^{w \cdot \lambda}(t) \right) \cdot \int_{\varphi \in \widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(\lambda)\otimes \varphi)}(\widehat{f}_0 , \dots, \widehat{f}_m)\cdot d\varphi\\ =& \sum_{\text{regular} \ \lambda \in \Lambda^{*}_{K\cap M_\circ}+ \rho_c} \left(\sum_{w \in W_{K \cap M_\circ}} (-1)^w \cdot e^{w \cdot \lambda}(t) \right) \cdot \int_{\varphi \in \widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(\lambda)\otimes \varphi)}(\widehat{f}_0 , \dots, \widehat{f}_m)\cdot d\varphi. \end{aligned} \end{align} $$

Here, the last equation follows from (4.10). Remembering that the above is anti-invariant under the $W_K$ -action, we can replace $\rho _c$ by $\rho _c^{M_\circ \cap K}$ . That is, (4.11) equals

$$\begin{align*}\sum_{\text{regular} \ \lambda \in \Lambda^{*}_{K\cap M_\circ}+ \rho_c^{M_\circ \cap K}} \left(\sum_{w \in W_{K \cap M_\circ}} (-1)^w \cdot e^{w \cdot \lambda}(t) \right) \cdot \int_{\varphi \in \widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(\lambda)\otimes \varphi)}(\widehat{f}_0, \dots, \widehat{f}_m)\cdot d\varphi. \end{align*}$$

Similarly, for the singular part,

$$\begin{align*}\begin{aligned} & \sum_{\text{singular} \ \lambda \in \Lambda^{*}_{K}+ \rho_c} \left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\lambda}= \sum_{\text{singular} \ \lambda \in \Lambda^{*}_{K \cap M_\circ}+ \rho_c^{M_\circ \cap K}} \left(\sum_{w \in W_{K \cap M_\circ}} (-1)^w \cdot e^{w \cdot \lambda}(t) \right)\\ & \times \left(\sum_{i=1}^{n(\lambda)} \frac{\epsilon(i)}{n(\lambda)} \cdot \int_{\varphi \in \widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}_i(\lambda)\otimes \varphi)}(\widehat{f}_0 , \dots, \widehat{f}_m) \right). \end{aligned} \end{align*}$$

Finally, for the higher part,

$$\begin{align*}\begin{aligned}\left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\mathrm{high}} &=\int_{\pi \in \widehat{G}^{\text{high}}_{\mathrm{temp}}} T_{\pi}(\widehat{f}_0 , \dots, \widehat{f}_m)\left( \sum_{\eta^{M_\circ} \otimes \varphi \in \mathcal{A}(\pi)} \kappa^{M_\circ} (\eta^{M_\circ}, t) \right) \cdot d\varphi\\ &= \int_{\eta^{M_\circ} \otimes \varphi \in \widehat{M}^{\text{high}}_{\mathrm{temp}} \times \widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\eta^{M_\circ} \otimes \varphi)}(\widehat{f}_0, \dots, \widehat{f}_m) \cdot \kappa^{M_\circ} (\eta^{M_\circ}, t). \end{aligned} \end{align*}$$

$\mathbf {Step \ 2} \colon $ We apply Proposition 4.10 and obtain the following.

• • regular part:

  $$\begin{align*}\begin{aligned} & \sum_{\text{regular} \ \lambda \in \Lambda^{*}_{K} + \rho_c} \left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\lambda}\\ &\quad =(-1)^m\sum_{\tau \in S_m}\mathrm{sgn}(\tau) \sum_{\text{regular} \ \lambda \in \Lambda^{*}_{K \cap M_\circ} + \rho_c^{M_\circ \cap K}} \left(\sum_{w \in W_{K \cap M_\circ}} (-1)^w \cdot e^{w \cdot \lambda}(t) \right)\\ &\qquad \int_{\varphi \in \widehat{A}_\circ} \int_{KM_\circ A_\circ N_\circ}\int_{G^{\times m} } H_{\tau(1)}\big(g_1 \dots g_m k\big)\dots H_{\tau(m)}\big(g_mk\big)\\ &\qquad e^{\langle \log \varphi+\rho, \log a\rangle} \cdot \Theta^{M_\circ}\big(\lambda\big)(m) \cdot f_0\big(kmank^{-1} (g_1 g_2 \dots g_m)^{-1}\big) f_1(g_1) \dots f_m(g_m). \end{aligned} \end{align*}$$

• • singular part:

  $$\begin{align*}\begin{aligned} & \sum_{\text{singular} \ \lambda \in \Lambda^{*}_{K} + \rho_c} \left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\lambda}\\ &\quad =(-1)^m\sum_{\tau \in S_m}\mathrm{sgn}(\tau) \sum_{\text{singular} \ \lambda \in \Lambda^{*}_{K \cap M_\circ} + \rho_c^{M_\circ \cap K}} \sum_{i=1}^{n(\lambda)} \left(\frac{\epsilon(i)}{n(\lambda)}\sum_{w \in W_{K \cap M_\circ}} (-1)^w \cdot e^{w \cdot \lambda}(t) \right)\\ &\qquad \int_{\varphi \in \widehat{A}_\circ} \int_{KM_\circ A_\circ N_\circ}\int_{G^{\times m} } H_{\tau(1)}\left(g_1 \dots g_m k\big)\dots H_{\tau(m)}\big(g_mk\right)\\ &\qquad e^{\langle \log \varphi+\rho, \log a\rangle} \cdot \Theta^{M_\circ}_i(\lambda)(m) \cdot f_0\left(kmank^{-1} (g_1 g_2 \dots g_m)^{-1}\right) f_1(g_1) \dots f_m(g_m). \end{aligned} \end{align*}$$

• • higher part:

  $$\begin{align*}\begin{aligned} &\left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\mathrm{high}}\\ &\quad =(-1)^m\int_{\eta^{M_\circ} \otimes \varphi \in \widehat{M}^{\text{high}}_{\mathrm{temp}} \times \widehat{A}_\circ} \int_{KM_\circ A_\circ N_\circ}\int_{G^{\times m} } H_{\tau(1)}\big(g_1 \dots g_m k\big)\dots H_{\tau(m)}\big(g_mk\big)\\ &\qquad e^{\langle \log \varphi+\rho, \log a\rangle} \cdot \Theta^{M_\circ}\big(\eta^{M_\circ}\big)(m) \cdot f_0\big(kmank^{-1} (g_1 g_2 \dots g_m)^{-1}\big)\\ &\qquad f_1(g_1) \dots f_m(g_m) \cdot \kappa^{M_\circ}(\eta^{M_\circ}, t). \end{aligned} \end{align*}$$

$\mathbf {Step \ 3} \colon $ All the above computations imply that

(4.12) $$ \begin{align} \begin{aligned} &\widehat{\Phi}_{t}(\widehat{f}_0 , \dots, \widehat{f}_m)\\ &\quad = \sum_{\text{regular} \ \lambda \in \Lambda^{*}_{K} + \rho_c} \left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\lambda} + \sum_{\text{singular} \ \lambda \in \Lambda^{*}_{K} + \rho_c} \left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\lambda} + \left[\widehat{\Phi}_{t}(\widehat{f}_0, \dots, \widehat{f}_m)\right]_{\mathrm{high}}\\ &\quad =(-1)^m \int_{K N_\circ}\int_{G^{\times m} } f' \cdot \left( \sum_{\tau \in S_m}\mathrm{sgn}(\tau) \cdot H_{\tau(1)}\big(g_1 \dots g_m k\big)\dots H_{\tau(m)}\big(g_mk\big)\cdot f_1(g_1) \dots f_m(g_m) \right),\\ \end{aligned} \end{align} $$

where

$$\begin{align*}\begin{aligned} &f' (t, k, n, g_1, \dots, g_m) \\ &\quad = \sum_{\text{regular} \ \lambda \in \Lambda^{*}_{K \cap M_\circ} + \rho_c^{M_\circ \cap K}} \left(\sum_{w \in W_{K \cap M_\circ}} (-1)^w \cdot e^{w \cdot \lambda}(t) \right) \cdot \int_{\varphi \in \widehat{A}_\circ} \left(\Theta^{M_\circ}(\lambda)\otimes \varphi\right)(c)\\ &\qquad + \sum_{\text{singular} \ \lambda \in \Lambda^{*}_{K \cap M_\circ} + \rho_c^{M_\circ \cap K}} \left(\frac{\sum_{w \in W_{K \cap M_\circ}} (-1)^w \cdot e^{w \cdot \lambda}(t) }{n(\lambda)}\right) \cdot \sum_{i=1}^{n(\lambda)} \epsilon(i) \cdot \int_{\varphi \in \widehat{A}_\circ} \left(\Theta^{M_\circ}_i(\lambda)\otimes \varphi\right)(c) \\ &\qquad + \int_{\eta^{M_\circ}\otimes \varphi \in \widehat{M}^{\text{high}}_{\mathrm{temp}} \otimes \widehat{A}_\circ} \left(\Theta^{M_\circ}(\eta^{M_\circ})\otimes \varphi \right)(c) \cdot \kappa^{M_\circ}(\eta^{M_\circ}, t), \end{aligned} \end{align*}$$

where

$$\begin{align*}c (k,m, a, n, g_1, \dots, g_m)= e^{\langle \rho, \log a \rangle }\cdot f_0\big(kmank^{-1} (g_1 g_2 \dots g_m)^{-1}\big). \end{align*}$$

Because T is a compact Cartan subgroup of $M_\circ $ , the sign function in the definition of Harish-Chandra’s orbital integral (1.1) is trivial [Reference Warner32, Section 8.1.1]. Hence, we apply Theorem C.7 to the function c and obtain

(4.13) $$ \begin{align} f' = F^T_c(t) = \Delta^{M_\circ}_T(t) \cdot \int_{h \in M_\circ/T_\circ} f_0\left(k h t h^{-1} n k^{-1} (g_1 g_2 \dots g_m)^{-1} \right). \end{align} $$

By (4.12) and (4.13), we conclude that

$$\begin{align*}\begin{aligned} &\widehat{\Phi}_{t}(\widehat{f}_0 , \dots, \widehat{f}_m)\\ &\quad = \Delta^{M_\circ}_T(t) \cdot (-1)^m\sum_{\tau \in S_m}\mathrm{sgn}(\tau) \int_{h \in M_\circ/T_\circ} \int_{K N_\circ}\int_{G^{\times m} } H_{\tau(1)}\big(g_1 \dots g_m k\big)\dots H_{\tau(m)}\big(g_mk\big)\\ &\qquad f_0\big(k hth^{-1} n k^{-1} (g_1 g_2 \dots g_m)^{-1} \big) f_1(g_1) \dots f_m(g_m)\\ &\quad = (-1)^m \Delta^{M_\circ}_T(t) \cdot \Phi_{P_\circ,t}(f_0, \dots, f_m). \end{aligned} \end{align*}$$

This completes the proof of Theorem 4.18.

5.1 Generators of $K_0(C^{*}_r(G))$

In Theorem C.4, we explain that the K-theory group of $C_r^{*}(G)$ is a free abelian group generated by the following components:

(5.1) $$ \begin{align} \begin{aligned} K_0(C^{*}_r(G))\cong& \bigoplus_{[P, \sigma]^{\mathrm{ess}}} K_0 \left(\mathcal{K}\left(C^{*}_r(G)_{[P, \sigma]}\right)\right)\\ \cong & \bigoplus_{ \lambda \in \Lambda^{*}_{K} +\rho_c} \mathbb{Z}. \end{aligned} \end{align} $$

Let $[P, \sigma ] \in \mathcal {P}(G)$ be an essential class corresponding to $\lambda \in \Lambda ^{*}_{K} + \rho _c$ . In this subsection, we construct a generator of $K_0(C^{*}_r(G))$ associated to $\lambda $ . We decompose $\widehat {A}_P = \widehat {A}_S \times \widehat {A}_\circ $ and denote $r = \dim \widehat {A}_S$ and $m=\dim \widehat {A}_\circ $ . Let V be an r-dimensional complex vector space and W an m-dimensional Euclidean space. Take

$$\begin{align*}z = (x_1, \cdots, x_r, y_1, \dots y_{m}), \quad x_i \in \mathbb{C}, y_j \in \mathbb{R} \end{align*}$$

to be coordinates on $V\oplus W$ . Assume that the finite group $(\mathbb {Z}_2)^{r}$ acts on V by simple reflections. In terms of coordinates,

$$\begin{align*}(x_1, \cdots, x_r, y_1, \dots, y_{m})\mapsto (\pm x_1, \cdots, \pm x_r, y_1, \dots, y_{m}). \end{align*}$$

Let us consider the Clifford algebra

$$\begin{align*}\mathrm{Clifford}(V) \otimes \mathrm{Clifford} (W) \end{align*}$$

together with the spinor module $S = S_{V} \otimes S_{W}$ . Here, the spinor modules are equipped with a $\mathbb {Z}_2$ -grading:

$$\begin{align*}S^+ = S_{V} ^+\otimes S_{W}^+ \oplus S_{V} ^-\otimes S_{W}^-, \quad S^- = S_{V} ^+\otimes S_{W}^- \oplus S_{V} ^-\otimes S_{W}^+. \end{align*}$$

Let $\mathcal {C}(V), \mathcal {C}(W)$ and $\mathcal {C}(V\oplus W)$ be the algebra of Schwartz functions on V, W and $V\oplus W$ . For any $z \in V \oplus W$ , the Clifford action $c(z) \colon S^\pm \to S^\mp $ is defined as follows.

Let $e_1, \dots e_{2^{r-1}}$ be a basis for $S^+_V$ , let $e_{2^{r-1}+1}, \dots e_{2^{r}}$ be a basis for $S^-_V$ , and let $f_1, \dots f_{2^{\frac {m}{2}}}$ be a basis for $S_W$ . We write

$$\begin{align*}c_{i, j, k, l}(z) = \langle c(z) e_i \otimes f_l, e_j \otimes f_k \rangle, \quad 1 \leq i, j \leq 2^r, 1 \leq k, l \leq 2^{\frac{m}{2}} \end{align*}$$

and define

$$\begin{align*}T:=\left( \begin{array}{cc} e^{-|z|^2} \cdot \mathrm{id}_{S^+}& e^{-\frac{|z|^2}{2}}(1-e^{-|z|^2})\cdot \frac{c(z)}{|z|^2}\\ e^{-\frac{|z|^2}{2}}c(z) & (1-e^{-|z|^2}) \cdot \mathrm{id}_{S^-} \end{array} \right) - \left(\begin{array}{cc} 0& 0\\ 0 & \mathrm{id}_{S^-} \end{array} \right) , \end{align*}$$

which is a $2^{r+\frac {m}{2}} \times 2^{r+\frac {m}{2}}$ matrix:

$$\begin{align*}\big(t_{i,j, k, l}\big), \quad 1 \leq i, j \leq 2^r, 1 \leq k, l \leq 2^{\frac{m}{2}}, \end{align*}$$

with $t_{i, j, k, l} \in \mathcal {C}(V \oplus W)$ .

Definition 5.1. On the m-dimensional Euclidean space W, we can define

$$\begin{align*}B^{m}=\left( \begin{array}{cc} e^{-|y|^2} \cdot \mathrm{id}_{S_{W} ^+}& e^{-\frac{|y|^2}{2}}(1-e^{-|y|^2})\cdot \frac{c(y)}{|y|^2}\\ e^{-\frac{|y|^2}{2}}c(y) & (1-e^{-|y|^2}) \cdot \mathrm{id}_{S_{W} ^-} \end{array} \right) - \left(\begin{array}{cc} 0& 0\\ 0 & \mathrm{id}_{S_{W}^-} \end{array} \right), \end{align*}$$

which is a $2^{\frac {m}{2}}\times 2^{\frac {m}{2}}$ matrix:

$$\begin{align*}\big(b_{k, l}\big), \quad 1 \leq k, l \leq 2^{\frac{m}{2}}, \end{align*}$$

with $b_{k, l} \in \mathcal {C}(W)$ . By straightforward computation, one can check that both the two matrices

$$\begin{align*}\left( \begin{array}{cc} e^{-|y|^2} \cdot \mathrm{id}_{S_{W} ^+}& e^{-\frac{|y|^2}{2}}(1-e^{-|y|^2})\cdot \frac{c(y)}{|y|^2}\\ e^{-\frac{|y|^2}{2}}c(y) & (1-e^{-|y|^2}) \cdot \mathrm{id}_{S_{W} ^-} \end{array} \right) , \quad\left(\begin{array}{cc} 0& 0\\ 0 & \mathrm{id}_{S_{W}^-}\end{array} \right) \end{align*}$$

are idempotents. In fact, $B^{m}$ is the Bott generator in $K_0(C_0(W)) \cong \mathbb {Z}$ .

Lemma 5.2. If we restrict to $W \subset V \oplus W$ (that is, $x = 0$ ), then

$$\begin{align*}T\big|_{x=0} = \left(\begin{array}{cc} \mathrm{id}_{S_{V}^+} & 0\\ 0 & -\mathrm{id}_{S_{V}^-} \end{array} \right) \otimes B^{m}. \end{align*}$$

Proof. By definition, we have that

• • $t_{i, j, k, l} = e^{-z^2}$ when $e_i, e_j, f_k, f_l \in S^+$ ;

• • $t_{i, j, k, l} = -e^{-z^2}$ when $e_i, e_j, f_k, f_l \in S^-$ ;

• • $t_{i, j, k, l} = e^{-\frac {|z|^2}{2}}(1-e^{-|z|^2})\cdot \frac {c_{i, j, k, l}(z)}{|z|^2}$ when $e_i, f_k, \in S^+$ and $e_j, f_l, \in S^-$ ;

• • $t_{i, j, k, l} = e^{-|z|^2} \cdot c_{i, j, k, l}(z)$ when $e_i, f_k, \in S^-$ and $e_j, f_l, \in S^+$ .

Moreover, the Clifford action $c(z)$ equals

$$\begin{align*}c(x) \otimes 1 + 1 \otimes c(y) \in \mathrm{End}(S_V) \otimes \mathrm{End}(S_W) \end{align*}$$

for $z = (x, y) \in V \oplus W$ . Thus, $c_{i,j, k, l}(z)\big |_{x = 0} = c_{k, l}(y)$ . This completes the proof.

Let $\sigma $ be a discrete series representation of $M_P$ and $\varphi \in \widehat {A}_\circ $ . Then,

$$\begin{align*}\varphi \otimes 1 \in \widehat{A}_\circ \times \widehat{A}_S = \widehat{A}_P. \end{align*}$$

Because $[P, \sigma ]$ is essential, the induced representation decomposes as

$$\begin{align*}\operatorname{\mathrm{Ind}}_{P}^G(\sigma \otimes \varphi \otimes 1) = \bigoplus_{i=1}^{2^r} \operatorname{\mathrm{Ind}}_{P_\circ}^G \big( \delta_i \otimes \varphi \big), \end{align*}$$

where $\delta _i$ are limit of discrete series representations of $M_\circ $ . By Equation (B.1), the characters of the limit of discrete series representations of $\delta _i$ are all the same up to a sign after restricting to a compact Cartan subgroup of $M_P$ . We can organize the numbering so that

$$\begin{align*}\delta_i, \quad i = 1, \dots 2^{r-1} \end{align*}$$

have the same character after restriction and

$$\begin{align*}\delta_i, \quad i = 2^{r-1}+1, \dots 2^{r} \end{align*}$$

have the same character. In particular, $\delta _i$ with $ 1 \leq i \leq 2^{r-1}$ and $\delta _j$ with $2^{r-1}+1\leq j \leq 2^r$ have the opposite characters after restriction.

We fix $2^r$ unit K-finite vectors $v_i \in \operatorname {\mathrm {Ind}}_{P}^G \big ( \delta _i \big )$ and define

(5.2) $$ \begin{align} S_\lambda := \big(t_{i,j, k, l}\cdot v_i \otimes v_j^{*}\big). \end{align} $$

The matrix

$$\begin{align*}S_\lambda \in \Big[\mathcal{C}\big(\widehat{A}_P, \mathcal{L}(\operatorname{\mathrm{Ind}}_P^G\sigma)\big) \Big]^{W_\sigma}, \end{align*}$$

and it is an idempotent. By the Morita equivalence (C.3),

$$\begin{align*}\mathcal{K}\big(\operatorname{\mathrm{Ind}}_P^G \sigma \big)^{W_\sigma} \sim \big(C_0(\mathbb{R})\rtimes \mathbb{Z}_2\big)^r \otimes C_0(\mathbb{R}^{m}). \end{align*}$$

Definition 5.3. We define

$$\begin{align*}Q_\lambda \in M_{2^{r+ m}}(\mathcal{C}(G)) \end{align*}$$

to be the wave packet associated to $S_\lambda $ . Then, $[Q_\lambda ]$ is the generator in $ K_0 \left (C^{*}_r(G)_{[P, \sigma ]}\right )$ for essential class $[P, \sigma ] \in \mathcal {P}(G)$ .

5.2 The main results

Let G be a linear connected real reductive Lie group with maximal compact subgroup K. We choose a maximal torus T of K, and $P_\circ $ a maximal cuspidal parabolic subgroup of G. It follows from Appendix C that for any $\lambda \in \Lambda ^{*}_{K} + \rho _c$ , there is a generator

$$\begin{align*}[Q_\lambda] \in K\big(C^{*}_r(G)\big). \end{align*}$$

In Section 3, we defined a family of cyclic cocycles

$$\begin{align*}\Phi_{P_\circ,e}, \hspace{5mm} \Phi_{P_\circ,t} \in HC\big(\mathcal{C}(G)\big) \end{align*}$$

for all $t\in T^{\text {reg}}$ and the maximal compact cuspidal parabolic subgroup $P_\circ $ .

Theorem 5.4. The index pairing between periodic cyclic cohomology and K-theory

$$\begin{align*}HP^{\text{even}}\big(\mathcal{C}(G)\big) \otimes K_0\big(\mathcal{C}(G)\big) \to \mathbb{C} \end{align*}$$

is given by

• • we have

  $$\begin{align*}\langle \Phi_{P_\circ,e}, [Q_\lambda] \rangle = \frac{1}{|W_{M_\circ \cap K}|} \cdot \sum_{w \in W_K} m\left(\sigma^{M_\circ}(w \cdot \lambda)\right), \end{align*}$$
  where $\sigma ^{M_\circ }(w \cdotp \lambda )$ is the discrete series representation with Harish-Chandra parameter $w \cdot \lambda $ , and $m\left (\sigma ^{M_\circ }(w \cdot \lambda )\right )$ is its Plancherel measure;

• • for any $t \in T^{\text {reg}}$ ,

  (5.3) $$ \begin{align} \langle \Phi_{P_\circ,t}, [Q_\lambda] \rangle = \frac{\sum_{w \in W_K} (-1)^w e^{w \cdot \lambda}(t)}{\Delta^{M_\circ}_T(t)}. \end{align} $$

The proof of Theorem 5.4 is presented in Sections 5.3 and 5.4.

Corollary 5.5. The index paring of $[Q_\lambda ]$ and normalized higher orbital integral equals the character of the representation $\operatorname {\mathrm {Ind}}_{P_\circ }^G(\sigma ^{M_\circ }(\lambda ) \otimes \varphi )$ at $\varphi = 1$ . That is,

$$\begin{align*}\left\langle \frac{\Delta^{M_\circ}_T}{\Delta^{G}_T}\cdot \Phi_{P_\circ,t}, [Q_\lambda] \right\rangle = \Theta(P_\circ, \sigma^{M_\circ}(\lambda), 1)(t). \end{align*}$$

Proof. It follows from applying the character formula, Corollary B.6, to the right side of Equation (5.3).

We notice that though the cocycles $\Phi _{P_\circ ,t}$ introduced in Definition 3.3 are only defined for regular elements in T, Theorem 5.4 suggests that the pairing $\Delta ^{M_\circ }_T(t) \langle \Phi _{P_\circ ,t}, [Q_\lambda ]\rangle $ is a well-defined smooth function on T. This inspires us to introduce the following map.

Definition 5.7. Define a map $\mathcal {F}^T\colon K_0(C^{*}_r(G))\to C^\infty (T)$ by

$$\begin{align*}\mathcal{F}^T([Q_\lambda])(t) \colon = \Delta^{M_\circ}_T \cdot \langle \Phi_{P_\circ,t}, [Q_\lambda]\rangle, \quad \lambda \in \Lambda_K^{*}+ \rho_c. \end{align*}$$

The map $\mathcal {F} ^T$ is first defined on the regular part $T^{\operatorname {\mathrm {reg}}}$ but can be extended smoothly to all elements in T as the right-hand side of the above equation extends to a smooth function on T.

By the Weyl character formula, for any irreducible K-representation $V_\lambda $ with highest weight $\lambda \in \Lambda _K^{*}$ , its character is given by

$$\begin{align*}\Theta_\lambda(t) = \frac{\sum_{w \in W_K} (-1)^w e^{w \cdot (\lambda+\rho_c)}(t)}{\Delta^{K}_T(t)}. \end{align*}$$

Multiplying by $\Delta ^K_T$ , we can identify $\text {Rep}(K)$ with the following subset of $C^\infty (T)$ :

$$\begin{align*}\left\{f \in C^\infty(T)\big| f(t) = \sum_{\lambda \in \Lambda^{*}_K + \rho_c} n_\lambda \cdot \left(\sum_{w \in W_K} (-1)^w e^{w \cdot (\lambda)}(t) \right), \quad n_\lambda \in \mathbb{Z} \right\}. \end{align*}$$

Under the above identification, we have the following corollary.

In [Reference Clare, Higson and Song5, Reference Clare, Higson, Song and Tang6], we use the above property of $\mathcal {F}^T$ to show that $\mathcal {F}^T$ is actually the inverse of the Connes-Kasparov Dirac index map, $\operatorname {\mathrm {index}}: \text {Rep}(K)\to K(C^{*}_r(G))$ .

Remark 5.9. The cyclic homology of the algebra $\mathcal {C}(G)$ was studied by Wassermann [Reference Wassermann34]. Wassermann’s result and the unpublished description of the decomposition of $\mathcal {C}(G)$ analogous to Equation (C.2) implies that the Connes-Chern character

$$\begin{align*}\operatorname{ch}: K_0\big(\mathcal{C}(G)\big)\to HP_{\text{even}}\big(\mathcal{C}(G)\big) \end{align*}$$

induces an isomorphism

$$\begin{align*}K_0\big(\mathcal{C}(G)\big)\otimes_{\mathbb{Z}} \mathbb{C}\cong HP_{\text{even}}\big(\mathcal{C}(G)\big). \end{align*}$$

Corollary 5.8 shows that higher orbital integrals $\Phi _{P_\circ ,t}, t\in T^{\text {reg}}$ distinguish $K_0(\mathcal {C}(G))$ . We can conclude from this fact that $\Phi _{P_\circ ,t}, t\in T^{\text {reg}}$ actually spans $HP^{\text {even}}\big (\mathcal {C}(G)\big )$ . As this outline of arguments involve some nontrivial unpublished works, we will not state this result as a ‘theorem’.

5.3 Regular case

Suppose that $\lambda \in \Lambda ^{*}_{K} + \rho _c$ is regular and $\sigma ^{M_\circ }(\lambda )$ is the discrete series representation of $M_\circ $ with Harish-Chandra parameter $\lambda $ . We consider the generator $[Q_\lambda ]$ , the wave packet associated to the matrix $S_\lambda $ introduced in (5.2), corresponding to

$$\begin{align*}\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(\lambda) \otimes \varphi), \hspace{5mm} \varphi \in \widehat{A}_\circ. \end{align*}$$

According to Theorem 4.15,

$$\begin{align*}\begin{aligned} (-1)^m\langle \Phi_{P_\circ,e}, Q_\lambda \rangle =&\int_{\pi \in \widehat{G}_{\mathrm{temp}}} T_{\pi}\left( \mathrm{Trace} \left(\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1} \right) \right)\cdot \mu(\pi ) \\ =& \int_{\widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(\lambda) \otimes \varphi) } \cdot \left( \mathrm{Trace} \left(\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1} \right) \right) \cdot \mu\left(\operatorname{\mathrm{Ind}}_{P}^G(\sigma^{M_\circ}(\lambda) \otimes \varphi)\right).\\ \end{aligned} \end{align*}$$

By Definition 4.13,

$$\begin{align*}\begin{aligned} \mu\left(\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(\lambda) \otimes \varphi)\right) &= \mu\left(\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(\lambda) \otimes 1\right) \\ &= \sum_{w \in W_K/ W_{K \cap M_\circ}} m\left(\sigma^{M_\circ}(w \cdot \lambda)\right)\\ &=\frac{1}{|W_{K\cap M_\circ}|} \cdot \sum_{w \in W_K} m\left(\sigma^{M_\circ}(w \cdot \lambda)\right). \end{aligned} \end{align*}$$

Moreover, in the case of regular $\lambda $ , $S_\lambda = [B^m \cdot (v \otimes v^{*})]$ , where $B^m$ is the Bott generator for $K_0(\mathcal {C}(\widehat {A}_\circ ))$ and v is a unit K-finite vector in $\operatorname {\mathrm {Ind}}_{P_\circ }^G(\sigma ^{M_\circ }(\lambda ))$ . By (4.2),

$$\begin{align*}\begin{aligned} &\int_{\widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma(\lambda) \otimes \varphi) }\left( \mathrm{Trace} \big(\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1} \big) \right) = \langle B^m, b_{m} \rangle = 1, \end{aligned} \end{align*}$$

where $[b_m] \in HC^m(\mathcal {C}(\mathbb {R}^m))$ is the cyclic cocycle on $\mathcal {C}(\mathbb {R}^m)$ of degree m; cf. Example 4.16. We conclude that

$$\begin{align*}\langle \Phi_{P_\circ,e}, [Q_\lambda] \rangle = \frac{(-1)^m}{|W_{K\cap M_\circ}|} \cdot \sum_{w \in W_K} m\left(\sigma^{M_\circ}(w \cdot \lambda)\right). \end{align*}$$

For the orbital integral $\Phi _{P_\circ ,t}$ , only the regular part will contribute. The computation is similar as above, and we conclude that

$$\begin{align*}\begin{aligned} \langle \Phi_{P_\circ,t}, Q_\lambda \rangle &= (-1)^m\sum_{w \in W_K} (-1)^w e^{w \cdot \lambda}(t) \cdot \int_{\widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma^{M_\circ}(\lambda) \otimes \varphi) }\left( \mathrm{Trace} \left(\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1} \right) \right)\\ &=(-1)^m \sum_{w \in W_K} (-1)^w e^{w \cdot \lambda}(t). \end{aligned} \end{align*}$$

5.4 Singular case

Suppose now that $\lambda \in \Lambda ^{*}_{K} + \rho _c$ is singular. We decompose

$$\begin{align*}\widehat{A}_{P} = \widehat{A}_\circ \times \widehat{A}_S, \hspace{5mm} \varphi = (\varphi_1, \varphi_2). \end{align*}$$

We denote $r = \dim (A_S)$ and $m = \dim (A_\circ )$ as before. In this case, we have that

$$\begin{align*}\operatorname{\mathrm{Ind}}_{P}^G(\sigma^M \otimes \varphi_1 \otimes 1) = \bigoplus_{i=1}^{2^r} \operatorname{\mathrm{Ind}}_{P_\circ}^G \big( \sigma^{M_\circ}_i \otimes \varphi_1 \big), \end{align*}$$

where $\sigma ^M$ is a discrete series representation of M and $ \sigma ^{M_\circ }_i $ , $i=1,..., 2^r$ , are limit of discrete series representations of $M_\circ $ with Harish-Chandra parameter $\lambda $ .

Recall that the generator $Q_\lambda $ is the wave packet associated to $S_\lambda $ . The index paring equals

$$\begin{align*}(-1)^m\langle \Phi_{P_\circ,e}, Q_\lambda \rangle = \int_{\widehat{A}_P} T_{\operatorname{\mathrm{Ind}}_{P}^G(\sigma^M\otimes \varphi)}\left( \mathrm{Trace} \left(\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1} \right) \right)\cdot \mu\left( \operatorname{\mathrm{Ind}}_{P}^G(\sigma^M \otimes \varphi) \right). \end{align*}$$

By the definition of $\mu $ ,

$$\begin{align*}\mu\left( \operatorname{\mathrm{Ind}}_{P}^G(\sigma^M \otimes \varphi) \right) = \sum_{\eta^{M_\circ} \otimes \varphi_1 \in \mathcal{A}\left(( \operatorname{\mathrm{Ind}}_{P}^G(\sigma^M \otimes \varphi) \right)} m(\eta^{M_\circ}). \end{align*}$$

Thus, the function $\mu \big ( \operatorname {\mathrm {Ind}}_{P}^G(\sigma ^M \otimes \varphi _1\otimes \varphi _2) \big )$ is constant with respect to $\varphi _1 \in \widehat {A}_\circ $ . It follows from (4.2) that

$$\begin{align*}\begin{aligned} &(-1)^m\langle \Phi_{P_\circ,e}, Q_\lambda \rangle \\ &\quad = \int_{\widehat{A}_S} \left(\mu\left( \operatorname{\mathrm{Ind}}_{P}^G(\sigma^M \otimes \varphi) \right) \cdot \int_{\widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P}^G(\sigma^M, \varphi)}\left( \mathrm{Trace} (\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1} ) \right)\right) \\ &\quad = \sum_{\tau \in S_m} \sum_{j_0 = i_1, \dots j_{m-1} = i_m, j_m = i_0} \\ &\qquad \int_{\widehat{A}_S} \left( \mu\big( \operatorname{\mathrm{Ind}}_{P}^G(\sigma^M \otimes \varphi) \int_{\widehat{A}_\circ} (-1)^\tau \cdot s_{i_0, j_0}(\varphi) \frac{\partial s_{i_1, j_1}(\varphi)}{\partial_{\tau(1)}} \dots \frac{\partial s_{i_m, j_m}(\varphi)}{\partial_{\tau(m)}} \right), \end{aligned} \end{align*}$$

where $s_{i, j} \in \mathcal {C}(\widehat {A}_\circ \times \widehat {A}_S)$ with $1 \leq i, j \leq 2^{r + \frac {m}{2}}$ is the coefficient of the $(i,j)$ -th entry in the matrix $S_\lambda $ . We notice that the dimension of $\widehat {A}_P$ is $m+r$ . It follows from the Connes-Hochschild-Kostant-Rosenberg theorem ([Reference Connes7, Theorem 46]) that the periodic cyclic cohomology of the algebra $\mathcal {C}(\widehat {A}_P)$ is spanned by a cyclic cocycle of degree $m+r$ . Accordingly, we conclude that

$$\begin{align*}\langle [\Phi_{P_\circ,e}], Q_\lambda \rangle = 0 \end{align*}$$

because it equals the pairing of the Bott element $B^{m+r} \in K(\widehat {A}_P)$ and a cyclic cocycle in $HC(\widehat {A}_P)$ with degree only $m<m+r$ .

Next, we turn to the index pairing of orbital integrals $\Phi _{P_\circ ,t}$ for $t \in T^{\operatorname {\mathrm {reg}}}$ . In this singular case, it is clear that the regular part of higher orbital integrals will not contribute. For the higher part,

$$\begin{align*}\begin{aligned} &(-1)^m\langle [\Phi_{P_\circ,t}]_{\text{high}}, Q_\lambda \rangle \\ &\quad = \int_{\varphi \in \widehat{A}_P} T_{\operatorname{\mathrm{Ind}}_{P}^G(\sigma^M \otimes \varphi)}\left( \mathrm{Trace} \left(\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1} \right) \right) \cdot \left( \sum_{\eta^{M_\circ} \otimes \varphi_1 \in \mathcal{A}(\operatorname{\mathrm{Ind}}_{P}^G(\sigma^M \otimes \varphi))} \kappa^{M_\circ} (\eta^{M_\circ}, t) \right). \end{aligned} \end{align*}$$

Note that the function

$$\begin{align*}\sum_{\eta^{M_\circ} \otimes \varphi_1 \in \mathcal{A}(\operatorname{\mathrm{Ind}}_{P}^G(\sigma^M \otimes \varphi))} \kappa^{M_\circ} (\eta^{M_\circ}, t) \end{align*}$$

is constant in $\varphi _1 \in \widehat {A}_\circ $ . By (4.2), we see that

$$\begin{align*}\begin{aligned} &\int_{\widehat{A}_S} \left( \sum_{\eta^{M_\circ} \otimes \varphi_1 \in \mathcal{A}\left(\operatorname{\mathrm{Ind}}_{P}^G\left(\sigma^M \otimes \varphi \right)\right)} \kappa^{M_\circ} \left(\eta^{M_\circ}, t\right) \cdot \int_{ \widehat{A}_\circ} T_{\operatorname{\mathrm{Ind}}_{P}^G\left(\sigma^M \otimes \varphi\right)}\mathrm{Trace} \left(\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1}\right) \right)\\ &\quad = \sum_{\tau \in S_m} \quad \sum_{\eta^{M_\circ} \otimes \varphi_1 \in \mathcal{A}(\operatorname{\mathrm{Ind}}_{P}^G(\sigma^M \otimes \varphi))} \quad \sum_{j_0 = i_1, \dots j_{m-1} = i_m, j_m = i_0} \\ &\qquad \int_{\widehat{A}_S} \int_{\widehat{A}_\circ} \Bigg( \kappa^{M_\circ} (\eta^{M_\circ}, t) (-1)^\tau \cdot s_{i_0, j_0}(\varphi) \frac{\partial s_{i_1, j_1}(\varphi)}{\partial_{\tau(1)}} \dots \frac{\partial s_{i_m, j_m}(\varphi)}{\partial_{\tau(m)}} \Bigg). \end{aligned} \end{align*}$$

We conclude that

$$\begin{align*}\langle [\Phi_{P_\circ,t}]_{\text{high}}, Q_\lambda \rangle = 0 \end{align*}$$

because it equals the paring of the Bott element $B^{m+r} \in K(\widehat {A}_P)$ and a cyclic cocycle on $\mathcal {C}(\widehat {A}_P)$ of degree m, which is trivial in $HP^{\text {even}}(\mathcal {C}(\widehat {A}_P))$ .

For the singular part, by the Schur’s orthogonality, we have $\langle [\Phi _{P_\circ ,t}]_{\lambda '}, Q_\lambda \rangle = 0$ unless $\lambda ' = \lambda $ . When $\lambda '=\lambda $ , Theorem 4.18 gives us the following computation:

$$\begin{align*}\begin{aligned} &(-1)^m\langle [\Phi_{P_\circ,t}]_{\lambda}, Q_\lambda \rangle\\ &\quad = \left( \frac{1}{2^r}\sum_{w \in W_K} (-1)^w e^{w \cdot \lambda}(t) \right) \cdot \sum_{k=1}^{2^r}\int_{\varphi \in \widehat{A}_P} \epsilon(k) \cdot T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma_k^{M_\circ} \otimes \varphi_1)}\left( \mathrm{Trace} \left(\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1} \right) \right). \end{aligned} \end{align*}$$

For each fixed k, it follows from Lemma 5.2 and Lemma 4.11 that

$$\begin{align*}\begin{aligned} &\int_{\varphi \in \widehat{A}_P} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma_k^{M_\circ} \otimes \varphi_1)}\left( \mathrm{Trace} \left(\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1} \right) \right)\\ &\quad =\int_{\varphi \in \widehat{A}_P} T_{\operatorname{\mathrm{Ind}}_{P_\circ}^G(\sigma_k^{M_\circ} \otimes \varphi_1)}\left( \mathrm{Trace} \left(\underbrace{S_\lambda \otimes \dots \otimes S_\lambda}_{m+1} \right)\Big|_{\varphi_2=0} \right)\\ &\quad = \sum_{j_0 = i_1, \dots j_{m-1} = i_m, j_m = i_0 = k} \sum_{\tau \in S_m}\int_{\varphi_1 \in \widehat{A}_\circ} (-1)^\tau \cdot s_{i_0, j_0}(\varphi_1) \frac{\partial s_{i_1, j_1}(\varphi_1)}{\partial_{\tau(1)}} \dots \frac{\partial s_{i_m, j_m}(\varphi_1)}{\partial_{\tau(m)}}\\ &\quad = \begin{cases} \langle B^m, b_m\rangle = 1 & \text{if} \ k = 1, \dots 2^{r-1} \\ -\langle B^m, b_m\rangle = 1 & \text{if} \ k = 2^{r-1}+ 1, \dots 2^{r}. \\ \end{cases} \end{aligned} \end{align*}$$

Combining all the above together and the fact that $\epsilon (k) = 1$ for $k = 1, \dots 2^{r-1}$ and $\epsilon (k) = -1$ for $k = 2^{r-1}+1, \dots 2^{r}$ , we conclude that

$$\begin{align*}\langle [\Phi_{P_\circ,t}], Q_\lambda \rangle = \langle [\Phi_{P_\circ,t}]_{\lambda}, Q_\lambda \rangle = (-1)^m \sum_{w \in W_K} (-1)^w e^{w \cdot \lambda}(t). \end{align*}$$