AI 中文总结
该研究针对实内形式K-理论，明确离散系列特征恒等式的积分结构，证明相关同态复合公式，推导正合阻碍序列，得到稳定K-理论格间的典范转移，并给出SL(2,R)及C_n型内形式的具体结果。
AI 中文摘要
设G为具有有限中心和离散系列的连通线性实半单群，G_c为固定的紧内形式。我们分离出离散系列特征恒等式背后的积分结构。存在自然同态$\boldsymbol{\textit{C}}_G:K_0(C_r^*(G))\to R(G_c)$与$\boldsymbol{\textit{J}}_G:R(G_c)\to K_0(C_r^*(G))$，分别由稳定和普通椭圆轨道积分刻画：前者将非紧Dolbeault-Dirac指标映射到其紧对应指标，后者将G_c的不可约表示映射到对应离散系列L-包中K-理论类的带符号和。我们证明$\boldsymbol{\textit{C}}_G\boldsymbol{\textit{J}}_G=[W_G:W_K]\boldsymbol{\text{id}}_{R(G_c)}$，因此包基数是拆分稳定轨道平均的精确积分代价。令$S_G=\text{im}\boldsymbol{\textit{J}}_G$、$U_G=\text{ker}\boldsymbol{\textit{C}}_G$，得到正合阻碍序列$0\to S_G\bigoplus U_G\to K_0(C_r^*(G))\to R(G_c)/[W_G:W_K]R(G_c)\to 0$。对包基数取逆后，这给出实内形式稳定K-理论格之间的典范稳定投影子与函子转移。对于$\boldsymbol{\text{SL}}(2,\boldsymbol{\text{R}})$，阻碍为$(\boldsymbol{\text{Z}}/2\boldsymbol{\text{Z}})[z+z^{-1}]$；对于C_n型内形式，$\boldsymbol{\text{Sp}}(2n,\boldsymbol{\text{R}})$的相关乘子为$2^n$，$\boldsymbol{\text{Sp}}(p,n-p)$的相关乘子为$\binom{n}{p}$。
英文摘要
Let $G$ be a connected linear real semisimple group with finite centre and discrete series, and let $G_c$ be a fixed compact inner form. We isolate an integral structure behind the discrete-series character identity. There are natural homomorphisms $\mathcal{C}_G:K_0(C_r^*(G))\longrightarrow R(G_c)$ and $\mathcal{J}_G:R(G_c)\longrightarrow K_0(C_r^*(G))$, characterised, respectively, by stable and ordinary elliptic orbital integrals. The first sends a noncompact Dolbeault--Dirac index to its compact counterpart; the second sends an irreducible representation of $G_c$ to the signed sum of the $K$-theory classes in the corresponding discrete-series $L$-packet. We prove $\mathcal{C}_G\mathcal{J}_G=[W_G:W_K]\,\mathrm{id}_{R(G_c)}$. Thus the packet cardinality is the precise integral cost of splitting stable orbital averaging. Writing $S_G=\operatorname{im}\mathcal{J}_G$ and $U_G=\ker\mathcal{C}_G$, we obtain the exact obstruction sequence $0\longrightarrow S_G\oplus U_G\longrightarrow K_0(C_r^*(G))\longrightarrow R(G_c)/[W_G:W_K]R(G_c)\longrightarrow 0$. After inverting the packet cardinality, this yields a canonical stable projector and functorial transfers between the stable $K$-theory lattices of real inner forms. For $\operatorname{SL}(2,\mathbb{R})$ the obstruction is $(\mathbb{Z}/2\mathbb{Z})[z+z^{-1}]$. For the inner forms of type $C_n$, the relevant multiplier is $2^n$ for $\operatorname{Sp}(2n,\mathbb{R})$ and $\binom{n}{p}$ for $\operatorname{Sp}(p,n-p)$.
