Secure Connection Failed when trying to access https://localhost

Hi everyone,

Windows Server 2008, IIS 7.0

Only one web site, the Default.

Bindings only on 443 with SSL certificate issued by GeoTrust

Public DNS redirects http to https, Firewall has http closed, web site has "Require SSL" checked, web site logic codded to redirect any http to https in Application_BeginRequest.

I tried with client certificates on Ignore and Accept.

I need to access the https://localhost internally.

Error: Secure Connection Failed, ... can not be shown because the authenticity of the received data could not be verified

Context: The Windows Task Scheduler runs regularly a task, that is running a vbs script that makes a GET ofhttps://localhost.

Problem is that GeoTrust certificate is issues for mydomain.com. How can make localhost be trusted internally, or how can I internally access my site via mydomain.com with request comming from ::1 ?

TESTED: modified the hosts file on the server with
127.0.0.1    mydomain.com       www.mydomain.com
::1             mydomain.com       www.mydomain.com

ipconfig /flushdns  ==> same error when trying to access www.mydomain.com that certainly has a valid certificate.

Any ideas?

Thank you,

Alex