mod_auth_cas redirects to nonexistent DirectoryIndex?

[Josh Kelley]

I'm having a very odd problem with mod_auth_cas...  The first time I
visit a CAS-protected web site (https://www/private/), it redirects me
to the CAS login page, as expected, and then redirects me back, with a
successful login.  However, after having logged in and gotten a
ticket, if I try to reload the web site or revisit it, it redirects me
to the CAS serviceValidate page again (I thought that mod_auth_cas set
a cookie to remove the need for this?) and passes a service of
https://www/private/index.php, rather than using a service of
https://www/private/ and letting Apache's DirectoryIndex directive
figure out the rest.

Even weirder is that only Firefox exhibits this behavior; Internet
Explorer works fine.

Any suggestions for tracking down this problem?

Josh Kelley