Let mathfrak g=operatornameso(2n+1) and let mathfrak h=operatornameso(2k+1)×operatornameso(2n-2k+1). Let Y and Z be regular semisimple elements, and suppose that there exists a regular semisimple X∈mathfrak g with P_X^0=P_Y^0P_Z^0. On the equal-valuation strip, let the transfer factor be q^c operatornamesign(X,Y,Z), where c=c(n,k,r) and operatornamesign(X,Y,Z)∈0,1,-1. Langlands–Shelstad fundamental lemma. For all such Y and Z, q^c∑_X int_O(X)capmathfrak g(O_F)operatornamesign(X,Y,Z)=int_O^st(Y)× O^st(Z)capmathfrak h(O_F)1. This is the stated special case of the p-adic fundamental lemma, with the measures requiring appropriate normalization; no resolution is given in the supplied text.

Langlands–Shelstad fundamental lemma for orthogonal Lie algebras

Let ${\mathfrak g}=\operatorname{so}(2n+1)$ and let ${\mathfrak h}=\operatorname{so}(2k+1)\times\operatorname{so}(2n-2k+1)$ . Let $Y$ and $Z$ be regular semisimple elements, and suppose that there exists a regular semisimple $X\in{\mathfrak g}$ with $P_X^0=P_Y^0P_Z^0$ . On the equal-valuation strip, let the transfer factor be $q^c\operatorname{sign}(X,Y,Z)$ , where $c=c(n,k,r)$ and $\operatorname{sign}(X,Y,Z)\in\{0,1,-1\}$ . Langlands–Shelstad fundamental lemma. For all such $Y$ and $Z$ ,

q^c\sum_X\int_{O(X)\cap{\mathfrak g}(O_F)}\operatorname{sign}(X,Y,Z)=\int_{O^{st}(Y)\times O^{st}(Z)\cap{\mathfrak h}(O_F)}1.

This is the stated special case of the $p$ -adic fundamental lemma, with the measures requiring appropriate normalization; no resolution is given in the supplied text.

Sources & referencesView supporting material

Primary source

Thomas C. Hales, “Can p-adic integrals be computed?”, arXiv:math/0205207 (2002).

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Progress summary

Refreshed 7 hours ago

Solved

Ngô’s published general proof settles this conjectural identity, with the usual hypotheses and normalization of measures and transfer factors.

This is the orthogonal Lie-algebra specialization of the Langlands–Shelstad fundamental lemma, asserting equality between a signed orbital-integral sum and a stable orbital integral. Langlands formulated the fundamental lemma in 1983; the general Lie-algebra theorem now covers reductive groups, including this endoscopic setting.

arxiv.org · numdam.org · en.wikipedia.org

Known results

Laumon and Ngô proved the unitary-group case before the general theorem.
Waldspurger’s 2006 reduction transferred the function-field Lie-algebra result to local fields.
Waldspurger’s 2008 reduction showed that the Lie-algebra lemma implies the corresponding group lemma.
Ngô’s 2010 work proved the general Lie-algebra fundamental lemma geometrically via the Hitchin fibration.

arxiv.org · numdam.org · en.wikipedia.org

2010 general proof

Ngô’s theorem establishes the equality of the relevant $\kappa$ -orbital integral and stable orbital integral, with the factor $q^{r}$ , for general reductive groups under standard residual-characteristic hypotheses. Thus the displayed $\operatorname{so}(2n+1)$ endoscopic formula is settled as a specialization, although the retrieved sources do not separately expand its exact sum-and-integral notation.

Current status (as of August 2026): The general Langlands–Shelstad fundamental lemma for Lie algebras is proved, so this orthogonal specialization is settled subject to the standard hypotheses and correct normalization of measures and transfer factors.

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