Dependencies

Let \(q\mid 2m+2\) and \(d=(2m+2)/q\). Then \(N_q(\ell )=r_q\) if and only if each folded class modulo \(q\) contains at most \(d\) of the \(\ell _i\), and each fixed class at most \(\lfloor d/2\rfloor \).

Let \(q\mid m\) or \(q\mid 2m+1\), let \(\gcd (p,q)=1\), and let \(\lambda \) be a dominant weight of \(\mathrm{Sp}(2m)\). Then \(\chi _\lambda (g_{p/q})\neq 0\) if and only if each non-fixed folded class modulo \(q\) contains at most \(2m/q\), respectively \((2m+1)/q\), of the entries of \(\ell =\lambda +\rho \), and the fixed classes at most \(m/q\), respectively \((2m+1-q)/(2q)\); these bounds are then attained. In that case

where \(\Delta _\lambda \) is the product of dimensions of Theorem B.

Let \(t=2m+2\), \(d=\gcd (k,t)\) and \(q=t/d\). As a multiset, the spectrum \(A=\{ \zeta ^{\pm jk}: j=1,\dots ,m\} \) of \(a_P^{\, k}\) on \(\mathbb C^{2m}\), with \(\zeta \) of order \(t\), satisfies

Let \(G\) be simple and simply connected, \(\lambda \) dominant, \(\ell =\lambda +\rho \), \(\theta =p/q\) in lowest terms, and \(N_q(x)=\# \{ \alpha {\gt}0:\ q\mid (x,\alpha )\} \), \(r_q=N_q(\rho )\). Then \(N_q(\ell )\ge r_q\), and

Work in \(\mathbb {Z}[X]/(X^q-1)\). For \(x=(x_1,\dots ,x_m)\in \mathbb {Z}^m\) put

Then

and, for \(q\mid 2m+2\) and every \(\lambda \) satisfying the capacities of Corollary 4.1, \(\mathscr {R}_{\lambda +\rho }=\mathscr {R}_\rho \).

Let \(\lambda \) be a dominant weight of \(\mathrm{Sp}(2m)\), \(k\ge 1\), and \(d,q,p\) as above.

1. \(N_q(\ell )\ge N_q(\rho )\), and \(\chi _\lambda (a_P^{\, k})\neq 0\) if and only if \(N_q(\ell )=N_q(\rho )\). Equivalently, each folded class modulo \(q\) contains at most \(d\) of the entries \(\ell _1,\dots ,\ell _m\), and each fixed class at most \(\lfloor d/2\rfloor \).

2. If \(\chi _\lambda (a_P^{\, k})\neq 0\), then

   \[ \chi _\lambda (a_P^{\, k})\; =\; (-1)^{E_{p,q}(\lambda )}\, \frac{Z_q(\lambda +\rho )}{Z_q(\rho )} . \]

Let \(\lambda \) be as in Theorem A with \(\chi _\lambda (a_P^{\, k})\neq 0\). Then

where \(n_c\) is the number of \(\ell _i\) in the class \(c\), the factor for \(q/2\) is omitted when \(q\) is odd, and \(\kappa (\lambda )=1\) if \(d\) is odd or if the class \(0\) holds fewer than \(d/2\) of the \(\ell _i\), and \(\kappa (\lambda )=2\) otherwise.

Let \(m\ge 2\) and \(q\ge 2\). The identity

holds for every dominant weight \(\lambda \) with \(N_q(\lambda +\rho )=N_q(\rho )\) and every \(p\) prime to \(q\) if and only if \(q\mid m\), \(q\mid 2m+1\), \(q\mid 2m+2\), or \(q=6\) and \(m\equiv 1\pmod6\). Otherwise there is such a \(\lambda \) for which it fails at \(p=1\) even up to sign.

Let \(m\ge 2\) and \(\gcd (p,q)=1\). Then \(\chi _\lambda (g_{1/q})\in \mathbb {Z}\) for every dominant weight \(\lambda \) of \(\mathrm{Sp}(2m)\) if and only if at least one of

holds; and in that case \(g_{p/q}\) is conjugate to \(g_{1/q}\) for every \(p\) coprime to \(q\), so no character value depends on the numerator.

Let \(m\ge 2\), let \(q\ge 2\) be one of the denominators of Proposition 8.1, and let \(\lambda \) be dominant with \(\chi _\lambda (g_{p/q})\ne 0\). Then

More precisely, \(c_\lambda =1\) if \(q\mid m\), \(q\mid 2m+1\), or \(q=6\) and \(m\equiv 1\pmod6\); \(c_\lambda =\kappa (\lambda )\in \{ 1,2\} \) if \(q\mid 2m+2\); \(c_\lambda \in \{ 1,2\} \) if \(q\ge 4\) is even, \(q\mid 2m\) and \(q\nmid m\); and \(c_\lambda \in \{ 1,2,3,4,6\} \) if \(q=6\) and \(m\equiv 4\pmod6\). In the last two cases every listed value occurs.

Let \(\lambda \) be a dominant weight of \(\mathrm{Sp}(2m)\) with \(N_q(\lambda +\rho )=r_q\). Then \(b-\rho _Y\) is a dominant weight of \(Y\), the element \(g^Y_{p/q}\) is regular, and

For \(v\in \{ 0,a,q-1\} \), that is, for \(q\mid m\), \(q\mid 2m+1\) or \(q\mid 2m+2\), the last factor is \(1\) for every such \(\lambda \); for \(q\mid 2m+2\) the theorem is Theorems A(2) and B.