Excess intersection computes the A1 Hecke coefficients

The source zero-section embedding has codimension $a_v$ and normal bundle ${\mathrm{pr}}_1^{\ast }{T}^{\ast }{X}_v$. Its pullback to $Z_v$ is the zero section $Y_v\hookrightarrow \mathrm{Tot}{\mathcal{K}}_v$, with normal bundle ${\mathcal{K}}_v$ of rank $k=v(w-v-1)$. The normal map is the fiber inclusion of common $A$ into $q^{\ast }{T}^{\ast }{X}_v$. There is a canonical exact sequence $0⟶{\mathcal{K}}_v⟶q^{\ast }{T}^{\ast }{X}_v⟶{\mathrm{\Omega }}_{Y_v\mathrm{/}{X}_{v+1}}⟶0.$ Indeed tangent vectors to an $r$-fiber move $S$ inside fixed $T$; restriction of cotangent forms is the surjection dual to $T_{Y_v\mathrm{/}{X}_{v+1}}\hookrightarrow q^{\ast }T{X}_v$. Its kernel consists exactly of $A:W\mathrm{/}S\to S$ annihilating $T\mathrm{/}S$, namely $\mathrm{Hom}(W\mathrm{/}T,S)$. Thus the excess has rank $a_v-k=v$. The excess formula [F2] gives the first intersection class, of dimension $\dim Y_v-v=a_{v+1}$.

For any quotient projective bundle $p:\mathbb{P}(B)\to T$ of rank $n+1$, put $h=c_1(\mathcal{O}(1))$. Its relative Euler sequence is $0⟶{\mathrm{\Omega }}_p⟶p^{\ast }B\otimes \mathcal{O}(-1)⟶\mathcal{O}⟶0.$ The middle map evaluates the tautological quotient after tensoring by its inverse line; its kernel is the relative cotangent bundle, as one checks on a chart where one quotient coordinate is $1$. Thus [F3] gives $c_n({\mathrm{\Omega }}_p)={\sum }_{i=0}^n(-1{)}^{n-i}(n+1-i)p^{\ast }{c}_i(B)h^{n-i}$: on the splitting tower the roots are $b_j-h$, and in choosing a monomial with $i$ roots $b_j$ the remaining $n-i$ choices can be made in $n+1-i$ ways. Pushforward kills every $i>0$ term, since its hyperplane exponent is below $n$, and sends $h^n$ to $1$. Consequently $p_{\ast }(c_n({\mathrm{\Omega }}_p)\cap [\mathbb{P}(B)])=(-1{)}^n(n+1)[T]$. This also covers $n=0$.

In the other direction the same restriction to tangent vectors of a $q$-fiber gives $0⟶{\mathcal{K}}_v⟶r^{\ast }{T}^{\ast }{X}_{v+1}⟶{\mathrm{\Omega }}_{Y_v\mathrm{/}{X}_v}⟶0.$ A tangent vector moves $T$ while fixing $S$; annihilating all these variations forces the cotangent endomorphism with kernel $T$ to have image in $S$. This is again ${\mathcal{K}}_v$. Its excess rank is $a_{v+1}-k=w-v-1$, giving the second intersection class of dimension $\dim Y_v-(w-v-1)=a_v$. Neither exact sequence presumes a global splitting.

Apply step 1.2 first to $r$, the quotient projective bundle of the rank-$(v+1)$ tautological bundle on $X_{v+1}$, with $n=v$; then apply it to $q$, the bundle of lines in $W\mathrm{/}\mathcal{S}$, equivalently the quotient projective bundle of $(W\mathrm{/}\mathcal{S}{)}^{\ast }$, with $n=w-v-1$. Steps 1.1–1.2 give exactly the two asserted coefficients. All maps in the normal and Euler sequences are canonical restrictions, evaluations or differentials of equivariant projections, hence are $\mathrm{GL}(W)$-equivariant. No equivariant Chern parameter has been specialized silently: all pushforward identities just used are ordinary Chow identities.