Equivariant elliptic-complex index and character

Let $T=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^r$, where $r\ge 0$ is an integer, be a compact torus and $M$ a compact smooth manifold without boundary. A smooth left $T$-action is a jointly smooth map $a:T\times M\to M$ with $a(e,x)=x$ and $a(st,x)=a(s,a(t,x))$. Let $E^0,\dots ,E^N$ be finite-rank complex vector bundles on $M$. For each $j<N$, let $d_j:C^{\mathrm{\infty }}(M,E^j)\to C^{\mathrm{\infty }}(M,E^{j+1})$ be a linear differential operator of finite order $m_j$, and put $d_{-1}=d_N=0$. In local coordinates and bundle frames, if $d_j={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m_j}{a}_{j,\alpha }(x){\mathrm{\partial }}_x^{\alpha }$, use the positive-frequency principal symbol $\sigma (d_j)(x,\xi )={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=m_j}{a}_{j,\alpha }(x)(i\xi {)}^{\alpha }$. The sequence is a complex when $d_{j+1}{d}_j=0$ for every $j$. It is elliptic when, for every $x\in M$ and every nonzero real cotangent vector $\xi \in T_x^{\ast }M$, the sequence of principal-symbol maps

$0⟶E_x^0\stackrel{\sigma (d_0)(x,\xi )}{\to }{E}_x^1\stackrel{\sigma (d_1)(x,\xi )}{\to }\cdots \stackrel{\sigma (d_{N-1})(x,\xi )}{\to }{E}_x^N⟶0$

is exact. Suppose each bundle has a $T$-linearization: a jointly smooth map $T\times E^j\to E^j$ whose restrictions ${\rho }_t^j(x):E_x^j\to E_{t\cdot x}^j$ are complex-linear isomorphisms, with ${\rho }_e^j(x)=I_{E_x^j}$ and ${\rho }_{st}^j(x)={\rho }_s^j(t\cdot x)\circ {\rho }_t^j(x)$. The induced section maps

$(U_t^{j}s)(x):={\rho }_t^j(t^{-1}\cdot x)(s(t^{-1}\cdot x))$

commute with $d_j$. Require the cohomology spaces $H^j(E^{\bullet }):=\ker d_j\mathrm{/}\mathrm{im}{d}_{j-1}$ to be finite-dimensional. The representation ring $R(T)$ is the Grothendieck ring of finite-dimensional continuous complex representations of $T$. Define

${\mathrm{Ind}}_T(E^{\bullet }):={\sum }_{j=0}^N(-1{)}^j[H^j(E^{\bullet })]\in R(T),{\chi }_{E^{\bullet }}(t):={\sum }_{j=0}^N(-1{)}^j\mathrm{tr}\text{\hspace{0.17em}⁣}\left(U_t^j{\mid }_{H^j(E^{\bullet })}\right)\mathrm{.}$

For a two-term complex $0\to C^{\mathrm{\infty }}(M,E^{+})\stackrel{D^{+}}{\to }{C}^{\mathrm{\infty }}(M,E^{-})\to 0$, this gives $[\ker D^{+}]-[\mathrm{coker}{D}^{+}]$. This definition concerns elliptic complexes, not transversally elliptic operators.

Zelin Yi, §§1–2: elliptic complexes, induced cohomology maps, and the Lefschetz character

Berline–Getzler–Vergne, Chapter 6 §6.1: the equivariant Dirac index as a virtual character in $R(G)$

J. L. Heitsch, §3: cohomological Lefschetz number for an elliptic complex

For a $T$-invariant operator on a $T$-manifold, transversally elliptic means its principal symbol is invertible away from the zero section after restriction to $T_T^{\ast }M:=\{(x,\xi ):\xi (X_M(x))=0\text{ for every }X\in \mathfrak{t}\}$, where $X_M(x):={\frac{d}{ds}\mid }_{s=0}{\exp }_T(sX)\cdot x$. A transversally elliptic operator can have an infinite-dimensional kernel; for a compact torus its irreducible characters can occur with finite multiplicity but infinitely many weights, so its index is generally a distribution on $T$, not an element of $R(T)$. It is therefore excluded from this definition.